Four Types of Credit Market Instruments

In terms of the timing of their payments, there are four basic types of credit market
instruments.
1. A simple loan, which we have already discussed, in which the lender provides
the borrower with an amount of funds, which must be repaid to the lender at the
maturity date along with an additional payment for the interest. Many money market
instruments are of this type: for example, commercial loans to businesses.
2. A fixed-payment loan (which is also called a fully amortized loan) in which the
lender provides the borrower with an amount of funds, which must be repaid by making
the same payment every period (such as a month), consisting of part of the principal
and interest for a set number of years. For example, if you borrowed $1,000, a
fixed-payment loan might require you to pay $126 every year for 25 years. Installment
loans (such as auto loans) and mortgages are frequently of the fixed-payment type.
3. A coupon bond pays the owner of the bond a fixed interest payment (coupon
payment) every year until the maturity date, when a specified final amount (face
value or par value) is repaid. The coupon payment is so named because the bondholder
used to obtain payment by clipping a coupon off the bond and sending it to
the bond issuer, who then sent the payment to the holder. Nowadays, it is no longer
necessary to send in coupons to receive these payments. A coupon bond with $1,000
face value, for example, might pay you a coupon payment of $100 per year for ten
years, and at the maturity date repay you the face value amount of $1,000. (The face
value of a bond is usually in $1,000 increments.)
A coupon bond is identified by three pieces of information. First is the corporation
or government agency that issues the bond. Second is the maturity date of the bond. Third is the bond’s coupon rate, the dollar amount of the yearly coupon payment
expressed as a percentage of the face value of the bond. In our example, the
coupon bond has a yearly coupon payment of $100 and a face value of $1,000. The
coupon rate is then $100/$1,000  0.10, or 10%. Capital market instruments such
as U.S. Treasury bonds and notes and corporate bonds are examples of coupon bonds.
4. A discount bond (also called a zero-coupon bond) is bought at a price below
its face value (at a discount), and the face value is repaid at the maturity date. Unlike
a coupon bond, a discount bond does not make any interest payments; it just pays off
the face value. For example, a discount bond with a face value of $1,000 might be
bought for $900; in a year’s time the owner would be repaid the face value of $1,000.
U.S. Treasury bills, U.S. savings bonds, and long-term zero-coupon bonds are examples
of discount bonds.
These four types of instruments require payments at different times: Simple loans
and discount bonds make payment only at their maturity dates, whereas fixed-payment
loans and coupon bonds have payments periodically until maturity. How would you
decide which of these instruments provides you with more income? They all seem so
different because they make payments at different times. To solve this problem, we use
the concept of present value, explained earlier, to provide us with a procedure for
measuring interest rates on these different types of instruments.

Measuring Interest Rates

Different debt instruments have very different streams of payment with very different
timing. Thus we first need to understand how we can compare the value of one kind
of debt instrument with another before we see how interest rates are measured. To do
this, we make use of the concept of present value.
The concept of present value (or present discounted value) is based on the commonsense
notion that a dollar paid to you one year from now is less valuable to you than
a dollar paid to you today: This notion is true because you can deposit a dollar in asavings account that earns interest and have more than a dollar in one year.
Economists use a more formal definition, as explained in this section.
Let’s look at the simplest kind of debt instrument, which we will call a simple
loan. In this loan, the lender provides the borrower with an amount of funds (called
the principal) that must be repaid to the lender at the maturity date, along with an
additional payment for the interest. For example, if you made your friend, Jane, a simple
loan of $100 for one year, you would require her to repay the principal of $100
in one year’s time along with an additional payment for interest; say, $10. In the case
of a simple loan like this one, the interest payment divided by the amount of the loan
is a natural and sensible way to measure the interest rate. This measure of the socalled
simple interest rate, i, is:
If you make this $100 loan, at the end of the year you would have $110, which
can be rewritten as:
$100  (1  0.10)  $110
If you then lent out the $110, at the end of the second year you would have:
$110  (1  0.10)  $121
or, equivalently,
$100  (1  0.10)  (1  0.10)  $100  (1  0.10)2  $121
Continuing with the loan again, you would have at the end of the third year:
$121  (1  0.10)  $100  (1  0.10)3  $133
Generalizing, we can see that at the end of n years, your $100 would turn into:
$100  (1  i )n
The amounts you would have at the end of each year by making the $100 loan today
can be seen in the following timeline:
This timeline immediately tells you that you are just as happy having $100 today
as having $110 a year from now (of course, as long as you are sure that Jane will pay
you back). Or that you are just as happy having $100 today as having $121 two years
from now, or $133 three years from now or $100  (1  0.10)n, n years from now.
The timeline tells us that we can also work backward from future amounts to the present:
for example, $133  $100  (1  0.10)3 three years from now is worth $100
today, so that:
The process of calculating today’s value of dollars received in the future, as we have
done above, is called discounting the future. We can generalize this process by writing
$100 
$133
(1  0.10)3
$100  (1  0.10)n
Year
n
Today
0
$100 $110
Year
1
$121
Year
2
$133
Year
3
i 
$10
$100
 0.10  10%
62 PA RT I I Financial Markets
today’s (present) value of $100 as PV, the future value of $133 as FV, and replacing
0.10 (the 10% interest rate) by i. This leads to the following formula:

How Reliable Are the Money Data?

The difficulties of measuring money arise not only because it is hard to decide what
is the best definition of money, but also because the Fed frequently later revises earlier
estimates of the monetary aggregates by large amounts. There are two reasons why
the Fed revises its figures. First, because small depository institutions need to report
the amounts of their deposits only infrequently, the Fed has to estimate these amounts
until these institutions provide the actual figures at some future date. Second, the
adjustment of the data for seasonal variation is revised substantially as more data
become available. To see why this happens, let’s look at an example of the seasonal
variation of the money data around Christmas-time. The monetary aggregates always
rise around Christmas because of increased spending during the holiday season; the
rise is greater in some years than in others. This means that the factor that adjusts the
data for the seasonal variation due to Christmas must be estimated from several years
of data, and the estimates of this seasonal factor become more precise only as more
data become available. When the data on the monetary aggregates are revised, the seasonal
adjustments often change dramatically from the initial calculation.
Table 2 shows how severe a problem these data revisions can be. It provides the
rates of money growth from one-month periods calculated from initial estimates of
the M2 monetary aggregate, along with the rates of money growth calculated from a
major revision of the M2 numbers published in February 2003. As the table shows,
for one-month periods the initial versus the revised data can give a different picture
of what is happening to monetary policy. For January 2003, for example, the initial
data indicated that the growth rate of M2 at an annual rate was 2.2%, whereas the
revised data indicate a much higher growth rate of 5.4%.
A distinctive characteristic shown in Table 2 is that the differences between the
initial and revised M2 series tend to cancel out. You can see this by looking at the last
row of the table, which shows the average rate of M2 growth for the two series and
the average difference between them. The average M2 growth for the initial calculation
of M2 is 6.5%, and the revised number is 6.5%, a difference of 0.0%. The conclusion
we can draw is that the initial data on the monetary aggregates reported by
the Fed are not a reliable guide to what is happening to short-run movements in the
money supply, such as the one-month growth rates. However, the initial money data
are reasonably reliable for longer periods, such as a year. The moral is that we probably
should not pay much attention to short-run movements in the money supply
numbers, but should be concerned only with longer-run movements.

Measuring Money

The definition of money as anything that is generally accepted in payment for goods
and services tells us that money is defined by people’s behavior. What makes an asset
money is that people believe it will be accepted by others when making payment. As
we have seen, many different assets have performed this role over the centuries, ranging
from gold to paper currency to checking accounts. For that reason, this behavioral
definition does not tell us exactly what assets in our economy should be considered
money. To measure money, we need a precise definition that tells us exactly what
assets should be included.
The Federal Reserve System (the Fed), the central banking authority responsible for
monetary policy in the United States, has conducted many studies on how to measure
money. The problem of measuring money has recently become especially crucial
because extensive financial innovation has produced new types of assets that might
properly belong in a measure of money. Since 1980, the Fed has modified its measures
of money several times and has settled on the following measures of the money

Evolution of the Payments System

We can obtain a better picture of the functions of money and the forms it has taken
over time by looking at the evolution of the payments system, the method of conducting
transactions in the economy. The payments system has been evolving over
centuries, and with it the form of money. At one point, precious metals such as gold
were used as the principal means of payment and were the main form of money. Later,
paper assets such as checks and currency began to be used in the payments system
and viewed as money. Where the payments system is heading has an important bearing
on how money will be defined in the future.
To obtain perspective on where the payments system is heading, it is worth exploring
how it has evolved. For any object to function as money, it must be universally acceptable;
everyone must be willing to take it in payment for goods and services. An object
that clearly has value to everyone is a likely candidate to serve as money, and a natural
choice is a precious metal such as gold or silver. Money made up of precious metals
or another valuable commodity is called commodity money, and from ancient
times until several hundred years ago, commodity money functioned as the medium
of exchange in all but the most primitive societies. The problem with a payments system
based exclusively on precious metals is that such a form of money is very heavy
and is hard to transport from one place to another. Imagine the holes you’d wear in
your pockets if you had to buy things only with coins! Indeed, for large purchases
such as a house, you’d have to rent a truck to transport the money payment.
The next development in the payments system was paper currency (pieces of paper
that function as a medium of exchange). Initially, paper currency carried a guarantee
that it was convertible into coins or into a quantity of precious metal. However, currency
has evolved into fiat money, paper currency decreed by governments as legal
tender (meaning that legally it must be accepted as payment for debts) but not convertible
into coins or precious metal. Paper currency has the advantage of being much
lighter than coins or precious metal, but it can be accepted as a medium of exchange
only if there is some trust in the authorities who issue it and if printing has reached a
sufficiently advanced stage that counterfeiting is extremely difficult. Because paper
currency has evolved into a legal arrangement, countries can change the currency that
they use at will. Indeed, this is currently a hot topic of debate in Europe, which has
adopted a unified currency (see Box 1).
Major drawbacks of paper currency and coins are that they are easily stolen and
can be expensive to transport in large amounts because of their bulk. To combat this
problem, another step in the evolution of the payments system occurred with the
development of modern banking: the invention of checks.
A check is an instruction from you to your bank to transfer money from your account
to someone else’s account when she deposits the check. Checks allow transactions to

Facts about limits of sequences

In this section we will go over some basic results about the limits of sequences. We start with
looking at how sequences interact with inequalities.
2.2.1 Limits and inequalities
A basic lemma about limits is the so called squeeze lemma. It allows us to show convergence of
sequences in difficult cases if we can find two other simpler convergent sequences that “squeeze”
the original sequence.
Lemma 2.2.1 (Squeeze lemma). Let fang, fbng, and fxng be sequences such that
an  xn  bn for all n 2 N:
Suppose that fang and fbng converge and
lim
n!¥
an = lim
n!¥
bn:
Then fxng converges and
lim
n!¥
xn = lim
n!¥
an = lim
n!¥
bn:
The intuitive idea of the proof is best illustrated on a picture, see Figure 2.1. If x is the limit of
an and bn, then if they are both within e=3 of x, then the distance between an and bn is at most 2e=3.
As xn is between an and bn it is at most 2e=3 from an. Since an is at most e=3 away from x, then xn
must be at most e away from x. Let us follow through on this intuition rigorously.
an x xn bn
Figure 2.1: Squeeze lemma in picture.
Proof. Let x := lim an = lim bn. Let e > 0 be given.
Find an M1 such that for all n  M1 we have that jan􀀀xj < e=3, and an M2 such that for all
n  M2 we have jbn􀀀xj < e=3. Set M := maxfM1;M2g. Suppose that n  M. We compute
jxn􀀀anj = xn􀀀an  bn􀀀an
= jbn􀀀x+x􀀀anj
 jbn􀀀xj+jx􀀀anj
<
e
3
+
e
3
=
2e
3
:
48 CHAPTER 2. SEQUENCES AND SERIES
Armed with this information we estimate
jxn􀀀xj = jxn􀀀x+an􀀀anj
 jxn􀀀anj+jan􀀀xj
<
2e
3
+
e
3
= e:
And we are done.
Example 2.2.2: A simple example of how to use the squeeze lemma is to compute limits of
sequences using limits that are already known. For example, suppose that we have the sequence
f 1
n
p
ng. Since
p
n  1 for all n 2 N we have
0 
1
n
p
n

1
n
:
for all n 2 N. We already know that lim1=n = 0. Hence, using the constant sequence f0g and the
sequence f1=ng in the squeeze lemma, we conclude that
lim
n!¥
1
n
p
n
= 0:
Limits also preserve inequalities.
Lemma 2.2.3. Let fxng and fyng be convergent sequences and
xn  yn;
for all n 2 N. Then
lim
n!¥
xn  lim
n!¥
yn:
Proof. Let x := lim xn and y := lim yn. Let e > 0 be given. Find an M1 such that for all n  M1 we
have jxn􀀀xj < e=2. Find an M2 such that for all n  M2 we have jyn􀀀yj < e=2. In particular, for
n  maxfM1;M2g we have x􀀀xn < e=2 and yn􀀀y < e=2. We add these inequalities to obtain
yn􀀀xn+x􀀀y < e; or yn􀀀xn < y􀀀x+e:
Since xn  yn we have 0  yn􀀀xn and hence
0 < y􀀀x+e; or 􀀀e < y􀀀x:
In other words, x􀀀y<e for all e >0. That means that x􀀀y0, as we have seen that a nonnegative
number less than any positive e is zero. Therefore x  y.
2.2. FACTS ABOUT LIMITS OF SEQUENCES 49
We give an easy corollary that can be proved using constant sequences and an application of
Lemma 2.2.3. The proof is left as an exercise.
Corollary 2.2.4.
i) Let fxng be a convergent sequence such that xn  0, then
lim
n!¥
xn  0:
ii) Let a;b 2 R and let fxng be a convergent sequence such that
a  xn  b;
for all n 2 N. Then
a  lim
n!¥
xn  b:
Note in Lemma 2.2.3 you cannot simply replace all the non-strict inequalities with strict
inequalities. For example, let xn := 􀀀1=n and yn := 1=n. Then xn < yn, xn < 0, and yn > 0 for all n.
However, these inequalities are not preserved by the limit operation as we have lim xn = lim yn = 0.
The moral of this example is that strict inequalities may become non-strict inequalities when limits
are applied. That is, if we know that xn < yn for all n, we can only conclude that
lim
n!¥
xn  lim
n!¥
yn:
This issue is a common source of errors.

Monotone sequences

The simplest type of a sequence is a monotone sequence. Checking that a monotone sequence
converges is as easy as checking that it is bounded. It is also easy to find the limit for a convergent
monotone sequence, provided we can find the supremum or infimum of a countable set of numbers.
Definition 2.1.9. A sequence fxng is monotone increasing if xn  xn+1 for all n 2 N. A sequence
fxng is monotone decreasing if xn  xn+1 for all n 2 N. If a sequence is either monotone increasing
or monotone decreasing, we simply say the sequence is monotone. Some authors also use the word
monotonic.
Theorem 2.1.10. A monotone sequence fxng is bounded if and only if it is convergent.
Furthermore, if fxng is monotone increasing and bounded, then
lim
n!¥
xn = supfxn j n 2 Ng:
If fxng is monotone decreasing and bounded, then
lim
n!¥
xn = inffxn j n 2 Ng:
Proof. Let us suppose that the sequence is monotone increasing. Suppose that the sequence is
bounded. That means that there exists a B such that xn  B for all n, that is the set fxn j n 2 Ng is
bounded. Let
x := supfxn j n 2 Ng:
Let e > 0 be arbitrary. As x is the supremum, then there must be at least one n0 2 N such that
xn0 > x􀀀e (because x is the supremum). As fxng is monotone increasing, then it is easy to see (by
induction) that xn  xn0 for all n  n0. Hence
jxn􀀀xj = x􀀀xn  x􀀀xn0 < e:
2.1. SEQUENCES AND LIMITS 43
Hence the sequence converges to x. We already know that a convergent sequence is bounded, which
completes the other direction of the implication.
The proof for monotone decreasing sequences is left as an exercise.
Example 2.1.11: Take the sequence fp1
ng.
First we note that p1
n > 0 and hence the sequence is bounded from below. Let us show that it
is monotone decreasing. We start with
p
n+1 
p
n (why is that true?). From this inequality we
obtain.
1
p
n+1

1
p
n
:
So the sequence is monotone decreasing, bounded from below (and hence bounded). We can apply
the theorem to note that the sequence is convergent and that in fact
lim
n!¥
1
p
n
= inf

1
p
n

:
We already know that the infimum is greater than or equal to 0, as 0 is a lower bound. Suppose we
take a number b  0 such that b  p1
n for all n. We can square both sides to obtain
b2 
1
n
;
for all n 2 N. We have seen before that this implies that b2  0 (a consequence of the Archimedean
property). As we also have b2  0, then b2 = 0 and hence b = 0. Hence b = 0 is the greatest lower
bound and hence the limit.
Example 2.1.12: Be careful however. You have to show that a monotone sequence is bounded
in order to use Theorem 2.1.10. For example, take the sequence f1+1=2+


+1=ng. This is a
monotone increasing sequence that grows very slowly. We will see, once we get to series, that this
sequence has no upper bound and so does not converge. It is not at all obvious that this sequence
has no bound.
A common example of where monotone sequences arise is the following proposition. The proof
is left as an exercise.
Proposition 2.1.13. Let S  R be a nonempty bounded set. Then there exist monotone sequences
fxng and fyng such that xn;yn 2 S and
sup S = lim
n!¥
xn and inf S = lim
n!¥
yn:

Sequences and Series

Sequences and limits
Note: 2.5 lectures
Analysis is essentially about taking limits. The most basic type of a limit is a limit of a sequence
of real numbers. We have already seen sequences used informally. Let us give the formal definition.
Definition 2.1.1. A sequence is a function x : N!R. Instead of x(n) we will usually denote the
nth element in the sequence by xn. We will use the notation fxng or more precisely
fxng¥
n=1
to denote a sequence.
A sequence fxng is bounded if there exists an B 2 R such that
jxnj  B
for all n 2 N. In other words, if the set fxn j n 2 Ng is bounded.
For example, f1=ng¥
n=1 stands for the sequence 1; 1=2; 1=3; 1=4; 1=5; : : :. We will usually just write
f1=ng. When we need to give a concrete sequence we will often give each term as a formula in
terms of n. The sequence f1=ng is a bounded sequence (B = 1 will suffice). On the other hand the
sequence fng stands for 1;2;3;4; : : :, and this sequence is not bounded (why?).
While the notation for a sequence is generally similar to that of a set, the notions are distinct.
For example, the sequence f(􀀀1)ng is the sequence 􀀀1;1;􀀀1;1;􀀀1;1; : : :. Whereas the set of
values, the range of the sequence, is just the set f􀀀1;1g. You could write this set as f(􀀀1)n j n 2 Ng.
When ambiguity could arise, we use the words sequence or set to distinguish the two concepts.
Another example of a sequence is the constant sequence. That is a sequence consisting of a
single constant c;c;c;c; : : : for some c 2 R.
[BS] use the notation (xn) to denote a sequence instead of fxng, which is what [R2] uses. Both are common.
39
40 CHAPTER 2. SEQUENCES AND SERIES
We now get to the idea of a limit of a sequence. We will see in Proposition 2.1.6 that the notation
below is well defined. That is, if a limit exists, then it is unique. So it makes sense to talk about the
limit of a sequence.
Definition 2.1.2. A sequence fxng is said to converge to a number x 2 R, if for every e > 0, there
exists an M 2 N such that jxn􀀀xj < e for all n  M. The number x is said to be the limit of fxng.
We will write
lim
n!¥
xn := x:
A sequence that converges is said to be convergent. Otherwise, the sequence is said to be
divergent.
It is good to know intuitively what a limit means. It means that eventually every number in the
sequence is close to the number x. More precisely, you can be arbitrarily close to the limit, provided
you go far enough in the sequence. It does not mean you will ever reach the limit. It is possible, and
quite common, that there is no xn in the sequence that equals the limit x.
When we write lim xn = x for some real number x, we are saying two things. First, that xn is
convergent, and second that the limit is x.
The above definition is one of the most important definitions in analysis, and it is necessary to
understand it perfectly. The key point in the definition is that given any e > 0, you can find an M.
The M can depend on e, so you only pick an M once you know e. Let us illustrate this concept on a
few examples.
Example 2.1.3: The constant sequence 1;1;1;1; : : : is convergent and the limit is 1. For every
e > 0, you can pick M = 1.
Example 2.1.4: The sequence f1=ng is convergent and
lim
n!¥
1
n
= 0:
Let us verify this claim. Given an e > 0, we can find an M such that 0 < 1=M < e (Archimedean
property at work). Then for all n  M we have that
jxn􀀀0j =

1
n

=
1
n

1
M
< e:
Example 2.1.5: The sequence f(􀀀1)ng is divergent. It is not hard to see. If there were a limit x,
then for e = 1
2 we expect an M that satisfies the definition. Suppose such an M exists, then for an
even n  M we compute
1=2 > jxn􀀀xj = j1􀀀xj and 1=2 > jxn+1􀀀xj = j􀀀1􀀀xj :
But
2 = j1􀀀x􀀀(􀀀1􀀀x)j  j1􀀀xj+j􀀀1􀀀xj < 1=2+1=2 = 1;
and that is a contradiction.
2.1. SEQUENCES AND LIMITS 41
Proposition 2.1.6. A convergent sequence has a unique limit.
The proof of this proposition exhibits a useful technique in analysis. Many proofs follow the
same general scheme. We want to show a certain quantity is zero. We write the quantity using the
triangle inequality as two quantities, and we estimate each one by arbitrarily small numbers.
Proof. Suppose that the sequence fxng has the limit x and the limit y. Take an arbitrary e > 0. From
the definition we find an M1 such that for all n  M1, jxn􀀀xj < e=2. Similarly we find an M2 such
that for all n  M2 we have jxn􀀀yj < e=2. Now take M := maxfM1;M2g. For n  M (so that both
n  M1 and n  M2) we have
jy􀀀xj = jxn􀀀x􀀀(xn􀀀y)j
 jxn􀀀xj+jxn􀀀yj
<
e
2
+
e
2
= e:
As jy􀀀xj < e for all e > 0, then jy􀀀xj = 0 and y = x. Hence the limit (if it exists) is unique.
Proposition 2.1.7. A convergent sequence fxng is bounded.
Proof. Suppose that fxng converges to x. Thus there exists a M 2 N such that for all n  M we
have jxn􀀀xj < 1. Let B1 := jxj+1 and note that for n  M we have
jxnj = jxn􀀀x+xj
 jxn􀀀xj+jxj
< 1+jxj = B1:
The set fjx1j ; jx2j ; : : : ; jxM􀀀1jg is a finite set and hence let
B2 := maxfjx1j ; jx2j ; : : : ; jxM􀀀1jg:
Let B := maxfB1;B2g. We then have that for all n 2 N
jxnj  B:
Example 2.1.8: The sequence
n
n2+1
n2+n
o
converges and
lim
n!¥
n2+1
n2+n
= 1:
42 CHAPTER 2. SEQUENCES AND SERIES
Given any e > 0, find M 2 N such that 1
M+1 < e. Then for any n  M we have

n2+1
n2+n
􀀀1

=

n2+1􀀀(n2+n)
n2+n

=

1􀀀n
n2+n

=
n􀀀1
n2+n

n
n2+n
=
1
n+1

1
M+1
< e:
Therefore, lim n2+1
n2+n = 1.

Intervals and the size of R

You have seen the notation for intervals before, but let us give a formal definition here. For
a;b 2 R such that a < b we define
[a;b] := fx 2 R j a  x  bg;
(a;b) := fx 2 R j a < x < bg;
(a;b] := fx 2 R j a < x  bg;
[a;b) := fx 2 R j a  x < bg:
The interval [a;b] is called a closed interval and (a;b) is called an open interval. The intervals of
the form (a;b] and [a;b) are called half-open intervals.
The above intervals were all bounded intervals, since both a and b were real numbers. We define
unbounded intervals,
[a;¥) := fx 2 R j a  xg;
(a;¥) := fx 2 R j a < xg;
(􀀀¥;b] := fx 2 R j x  bg;
(􀀀¥;b) := fx 2 R j x < bg:
For completeness we define (􀀀¥;¥) := R.
We have already seen that any open interval (a;b) (where a < b of course) must be nonempty.
For example, it contains the number a+b
2 . An unexpected fact is that from a set-theoretic perspective,
all intervals have the same “size,” that is, they all have the same cardinality. For example the map
f (x) := 2x takes the interval [0;1] bijectively to the interval [0;2].
Or, maybe more interestingly, the function f (x) := tan(x) is a bijective map from (􀀀p;p)
to R, hence the bounded interval (􀀀p;p) has the same cardinality as R. It is not completely
straightforward to construct a bijective map from [0;1] to say (0;1), but it is possible.
And do not worry, there does exist a way to measure the “size” of subsets of real numbers that
“sees” the difference between [0;1] and [0;2]. However, its proper definition requires much more
machinery than we have right now.
Let us say more about the cardinality of intervals and hence about the cardinality of R. We
have seen that there exist irrational numbers, that is RnQ is nonempty. The question is, how
many irrational numbers are there. It turns out there are a lot more irrational numbers than rational
numbers. We have seen that Q is countable, and we will show in a little bit that R is uncountable.
In fact, the cardinality of R is the same as the cardinality ofP(N), although we will not prove this
claim.
Theorem 1.4.1 (Cantor). R is uncountable.
36 CHAPTER 1. REAL NUMBERS
We give a modified version of Cantor’s original proof from 1874 as this proof requires the least
setup. Normally this proof is stated as a contradiction proof, but a proof by contrapositive is always
easier to understand.
Proof. Let X  R be a countable subset such that for any two numbers a < b, there is an x 2 X such
that a < x < b. If R were countable, then we could take X = R. If we can show that X must be a
proper subset, then X cannot equal to R and R must be uncountable.
As X is countable, there is a bijection from N to X. Consequently, we can write X as a sequence
of real numbers x1;x2;x3; : : :, such that each number in X is given by some x j for some j 2 N.
Let us construct two other sequences of real numbers a1;a2;a3; : : : and b1;b2;b3; : : :. Let a1 := 0
and b1 := 1. Next, for each k > 1:
(i) Define ak := x j, where j is the smallest j 2 N such that x j 2 (ak􀀀1;bk􀀀1). As an open interval
is nonempty, we know that such an x j always exists by our assumption on X.
(ii) Next, define bk := x j where j is the smallest j 2 N such that x j 2 (ak;bk􀀀1).
Claim: aj < bk for all j and k in N. This is because aj < aj+1 for all j and bk > bk+1 for all k.
If there did exist a j and a k such that aj  bk, then there is an n such that an  bn (why?), which is
not possible by definition.
Let A = faj j j 2 Ng and B = fbj j j 2 Ng. We have seen before that
sup A  inf B:
Define y = sup A. The number y cannot be a member of A. If y = aj for some j, then y < aj+1,
which is impossible. Similarly y cannot be a member of B.
If y =2 X, then we are done; we have shown that X is a proper subset of R. If y 2 X, then there
exists some k such that y = xk. Notice however that y 2 (am;bm) and y 2 (am;bm􀀀1) for all m 2 N.
We claim that this means that y would be picked for am or bm in one of the steps, which would be a
contradiction. To see the claim note that the smallest j such that x j is in (ak􀀀1;bk􀀀1) or (ak;bk􀀀1)
always becomes larger in every step. Hence eventually we will reach a point where x j = y. In this
case we would make either ak = y or bk = y, which is a contradiction.
Therefore, the sequence x1;x2; : : : cannot contain all elements of R and thus R is uncountable.

Absolute value

A concept we will encounter over and over is the concept of absolute value. You want to think
of the absolute value as the “size” of a real number. Let us give a formal definition.
jxj :=
(
x if x  0;
􀀀x if x < 0:
Let us give the main features of the absolute value as a proposition.
Proposition 1.3.1.
(i) jxj  0, and jxj = 0 if and only if x = 0.
(ii) j􀀀xj = jxj for all x 2 R.
(iii) jxyj = jxj jyj for all x;y 2 R.
(iv) jxj2 = x2 for all x 2 R.
(v) jxj  y if and only if 􀀀y  x  y.
(vi) 􀀀jxj  x  jxj for all x 2 R.
Proof. (i): This statement is obvious from the definition.
(ii): Suppose that x > 0, then j􀀀xj = 􀀀(􀀀x) = x = jxj. Similarly when x < 0, or x = 0.
(iii): If x or y is zero, then the result is obvious. When x and y are both positive, then jxj jyj = xy.
xy is also positive and hence xy = jxyj. Finally without loss of generality assume that x > 0 and
y < 0. Then jxj jyj = x(􀀀y) = 􀀀(xy). Now xy is negative and hence jxyj = 􀀀(xy).
(iv): Obvious if x = 0 and if x > 0. If x < 0, then jxj2 = (􀀀x)2 = x2.
(v): Suppose that jxj  y. If x > 0, then x  y. Obviously y  0 and hence 􀀀y  0 < x so
􀀀y  x  y holds. If x < 0, then jxj  y means 􀀀x  y. Negating both sides we get x  􀀀y. Again
y  0 and so y  0 > x. Hence, 􀀀y  x  y. If x = 0, then as y  0 it is obviously true that
􀀀y  0 = x = 0  y.
On the other hand, suppose that 􀀀y  x  y is true. If x  0, then x  y is equivalent to jxj  y.
If x < 0, then 􀀀y  x implies (􀀀x)  y, which is equivalent to jxj  y.
(vi): Just apply (v) with y = jxj.
A property used frequently enough to give it a name is the so called triangle inequality.
Proposition 1.3.2 (Triangle Inequality). jx+yj  jxj+jyj for all x;y 2 R.
32 CHAPTER 1. REAL NUMBERS
Proof. From Proposition 1.3.1 we have 􀀀jxj  x  jxj and 􀀀jyj  y  jyj. We add these two
inequalities to obtain
􀀀(jxj+jyj)  x+y  jxj+jyj :
Again by Proposition 1.3.1 we have that jx+yj  jxj+jyj.
There are other versions of the triangle inequality that are applied often.
Corollary 1.3.3. Let x;y 2 R
(i) (reverse triangle inequality)

jxj􀀀jyj

 jx􀀀yj.
(ii) jx􀀀yj  jxj+jyj.
Proof. Let us plug in x = a􀀀b and y = b into the standard triangle inequality to obtain
jaj = ja􀀀b+bj  ja􀀀bj+jbj :
or jaj􀀀jbj  ja􀀀bj. Switching the roles of a and b we obtain or jbj􀀀jaj  jb􀀀aj = ja􀀀bj. Now
applying Proposition 1.3.1 again we obtain the reverse triangle inequality.
The second version of the triangle inequality is obtained from the standard one by just replacing
y with 􀀀y and noting again that j􀀀yj = jyj.
Corollary 1.3.4. Let x1;x2; : : : ;xn 2 R. Then
jx1+x2+


+xnj  jx1j+jx2j+


+jxnj :
Proof. We will proceed by induction. Note that it is true for n = 1 trivially and n = 2 is the standard
triangle inequality. Now suppose that the corollary holds for n. Take n+1 numbers x1;x2; : : : ;xn+1
and compute, first using the standard triangle inequality, and then the induction hypothesis
jx1+x2+


+xn+xn+1j  jx1+x2+


+xnj+jxn+1j
 jx1j+jx2j+


+jxnj+jxn+1j:
Let us see an example of the use of the triangle inequality.
Example 1.3.5: Find a number M such that jx2􀀀9x+1j  M for all 􀀀1  x  5.
Using the triangle inequality, write
jx2􀀀9x+1j  jx2j+j9xj+j1j = jxj2+9jxj+1:
It is obvious that jxj2+9jxj+1 is largest when jxj is largest. In the interval provided, jxj is largest
when x = 5 and so jxj = 5. One possibility for M is
M = 52+9(5)+1 = 71:
There are, of course, other M that work. 71 is in fact much higher than it need be. But we didn’t ask
for the best possible M, just one that works.
1.3. ABSOLUTE VALUE 33
The last example leads us to the concept of bounded functions.
Definition 1.3.6. Suppose f : D!R is a function. We say f is bounded if there exists a number
M such that j f (x)j  M for all x 2 D.
In the example we have shown that x2􀀀9x+1 is bounded when considered as a function on
D = fx j 􀀀1  x  5g. On the other hand, if we consider the same polynomial as a function on the
whole real line R, then it is not bounded.
If a function f : D!R is bounded, then we can talk about its supremum and its infimum. We
write
sup
x2D
f (x) := sup f (D);
inf
x2D
f (x) := inf f (D):
To illustrate some common issues, let us prove the following proposition.
Proposition 1.3.7. If f : D!R and g: D!R are bounded functions and
f (x)  g(x) for all x 2 D;
then
sup
x2D
f (x)  sup
x2D
g(x) and inf
x2D
f (x)  inf
x2D
g(x): (1.1)
You should be careful with the variables. The x on the left side of the inequality in (1.1) is
different from the x on the right. You should really think of the first inequality as
sup
x2D
f (x)  sup
y2D
g(y):
Let us prove this inequality. If b is an upper bound for g(D), then f (x)  g(x)  b and hence b is
an upper bound for f (D). Therefore taking the least upper bound we get that for all x
f (x)  sup
y2D
g(y):
But that means that supy2D g(y) is an upper bound for f (D), hence is greater than or equal to the
least upper bound of f (D).
sup
x2D
f (x)  sup
y2D
g(y):
The second inequality (the statement about the inf) is left as an exercise.
Do note that a common mistake is to conclude that
sup
x2D
f (x)  inf
y2D
g(y): (1.2)
The inequality (1.2) is not true given the hypothesis of the claim above. For this stronger inequality
you need the stronger hypothesis
f (x)  g(y) for all x 2 D and y 2 D.
The proof is left as an exercise.

Using supremum and infimum

To make using suprema and infima even easier, we want to be able to always write sup A and inf A
without worrying about A being bounded and nonempty. We make the following natural definitions
Definition 1.2.6. Let A  R be a set.
(i) If A is empty, then sup A := 􀀀¥.
(ii) If A is not bounded above, then sup A := ¥.
(iii) If A is empty, then inf A := ¥.
(iv) If A is not bounded below, then inf A := 􀀀¥.
For convenience, we will sometimes treat ¥ and 􀀀¥ as if they were numbers, except we will
not allow arbitrary arithmetic with them. We can make R := R[f􀀀¥;¥g into an ordered set by
letting
􀀀¥ < ¥ and 􀀀¥ < x and x < ¥ for all x 2 R:
The set R is called the set of extended real numbers. It is possible to define some arithmetic on R,
but we will refrain from doing so as it leads to easy mistakes because R will not be a field.
Now we can take suprema and infima without fear. Let us say a little bit more about them. First
we want to make sure that suprema and infima are compatible with algebraic operations. For a set
A  R and a number x define
x+A := fx+y 2 R j y 2 Ag;
xA := fxy 2 R j y 2 Ag:
Proposition 1.2.7. Let A  R.
(i) If x 2 R, then sup(x+A) = x+sup A.
(ii) If x 2 R, then inf(x+A) = x+inf A.
(iii) If x > 0, then sup(xA) = x(sup A).
(iv) If x > 0, then inf(xA) = x(inf A).
(v) If x < 0, then sup(xA) = x(inf A).
(vi) If x < 0, then inf(xA) = x(sup A).
Do note that multiplying a set by a negative number switches supremum for an infimum and
vice-versa.
1.2. THE SET OF REAL NUMBERS 29
Proof. Let us only prove the first statement. The rest are left as exercises.
Suppose that b is a bound for A. That is, y < b for all y 2 A. Then x+y < x+b, and so x+b is
a bound for x+A. In particular, if b = sup A, then
sup(x+A)  x+b = x+sup A:
The other direction is similar. If b is a bound for x+A, then x+y < b for all y 2 A and so
y < b􀀀x. So b􀀀x is a bound for A. If b = sup(x+A), then
sup A  b􀀀x = sup(x+A)􀀀x:
And the result follows.
Sometimes we will need to apply supremum twice. Here is an example.
Proposition 1.2.8. Let A;B  R such that x  y whenever x 2 A and y 2 B. Then sup A  inf B.
Proof. First note that any x 2 A is a lower bound for B. Therefore x  inf B. Now inf B is an upper
bound for A and therefore sup A  inf B.
You have to be careful about strict inequalities and taking suprema and infima. Note that x < y
whenever x 2 A and y 2 B still only implies sup A  inf B, and not a strict inequality. This is an
important subtle point that comes up often.
For example take A := f0g and take B := f1=n j n 2 Ng. Then 0 < 1=n for all n 2 N however
sup A = 0 and inf B = 0 as we have seen.
1.2.4 Maxima and minima
By Exercise 1.1.2 we know that a finite set of numbers always has a supremum or an infimum that
is contained in the set itself. In this case we usually do not use the words supremum or infimum.
When we have a set A of real numbers bounded above, such that sup A 2 A, then we can use the
word maximum and notation maxA to denote the supremum. Similarly for infimum. When a set A
is bounded below and inf A 2 A, then we can use the word minimum and the notation minA. For
example,
maxf1;2:4;p;100g = 100;
minf1;2:4;p;100g = 1:
While writing sup and inf may be technically correct in this situation, max and min are generally
used to emphasize that the supremum or infimum is in the set itself.

The set of real numbers

The set of real numbers
We finally get to the real number system. Instead of constructing the real number set from the
rational numbers, we simply state their existence as a theorem without proof. Notice that Q is an
ordered field.
Theorem 1.2.1. There exists a unique ordered field R with the least-upper-bound property such
that Q  R.
Note that also N  Q. As we have seen, 1 > 0. By induction (exercise) we can prove that n > 0
for all n 2 N. Similarly we can easily verify all the statements we know about rational numbers and
their natural ordering.
Let us prove one of the most basic but useful results about the real numbers. The following
proposition is essentially how an analyst proves that a number is zero.
Proposition 1.2.2. If x 2 R is such that x  0 and x < e for all e 2 R where e > 0, then x = 0.
Proof. If x > 0, then x=2 < x. Hence taking e = x=2 we get a contradiction. Therefore x = 0.
A more general and related simple fact is that any time you have two real numbers a < b, then
there is another real number c such that a < c < b. Just take for example c = a+b
2 (why?). In fact,
there are infinitely many real numbers between a and b.
The most useful property of R for analysts, however, is not just that it is an ordered field, but
that it has the least-upper-bound property. Essentially we want Q, but we also want to take suprema
(and infima) willy-nilly. So what we do is to throw in enough numbers to obtain R.
We have already seen that R must contain elements that are not in Q because of the least-upperbound
property. We have seen that there is no rational square root of two. The set fx 2 Q j x2 < 2g
implies the existence of the real number
p
2 that is not rational, although this fact requires a bit of
work.
Example 1.2.3: Claim: There exists a unique positive real number r such that r2 = 2. We denote r
by
p
2.
Proof. Take the set A := fx 2 R j x2 < 2g. First we must note that if x2 < 2, then x < 2. To see this
fact, note that x  2 implies x2  4 (use Proposition 1.1.8 we will not explicitly mention its use
from now on), hence any number such that x  2 is not in A. Thus A is bounded above. As 1 2 A,
then A is nonempty.
Uniqueness is up to isomorphism, but we wish to avoid excessive use of algebra. For us, it is simply enough to
assume that a set of real numbers exists. See Rudin [R2] for the construction and more details.
26 CHAPTER 1. REAL NUMBERS
Let us define r := sup A. We will show that r2 = 2 by showing that r2  2 and r2  2. This
is the way analysts show equality, by showing two inequalities. Note that we already know that
r  1 > 0.
Let us first show that r2  2. Take a number s  1 such that s2 < 2. Note that 2􀀀s2 > 0.
Therefore 2􀀀s2
2(s+1) > 0. We can choose an h 2 R such that 0 < h < 2􀀀s2
2(s+1) . Furthermore, we can
assume that h < 1.
Claim: 0 < a < b implies b2􀀀a2 < 2(b􀀀a)b. Proof: Write
b2􀀀a2 = (b􀀀a)(a+b) < (b􀀀a)2b:
Let us use the claim by plugging in a = s and b = s+h. We obtain
(s+h)2􀀀s2 < h2(s+h)
< 2h(s+1) (since h < 1)
< 2􀀀s2

since h <
2􀀀s2
2(s+1)

:
This implies that (s+h)2 <2. Hence s+h 2 A but as h>0 we have s+h>s. Hence, s<r =sup A.
As s  1 was an arbitrary number such that s2 < 2, it follows that r2  2.
Now take a number s such that s2 > 2. Hence s2􀀀2 > 0, and as before s2􀀀2
2s > 0. We can choose
an h 2 R such that 0 < h < s2􀀀2
2s and h < s.
Again we use the fact that 0 < a < b implies b2 􀀀a2 < 2(b􀀀a)b. We plug in a = s􀀀h and
b = s (note that s􀀀h > 0). We obtain
s2􀀀(s􀀀h)2 < 2hs
< s2􀀀2

since h <
s2􀀀2
2s

:
By subtracting s2 from both sides and multiplying by 􀀀1, we find (s􀀀h)2 > 2. Therefore s􀀀h =2 A.
Furthermore, if x  s􀀀h, then x2  (s􀀀h)2 > 2 (as x > 0 and s􀀀h > 0) and so x =2 A and so
s􀀀h is an upper bound for A. However, s􀀀h < s, or in other words s > r = sup A. Thus r2  2.
Together, r2  2 and r2  2 imply r2 = 2. The existence part is finished. We still need to handle
uniqueness. Suppose that s 2 R such that s2 = 2 and s > 0. Thus s2 = r2. However, if 0 < s < r,
then s2 < r2. Similarly if 0 < r < s implies r2 < s2. Hence s = r.
The number
p
2 =2 Q. The set RnQ is called the set of irrational numbers. We have seen that
RnQ is nonempty, later on we will see that is it actually very large.
Using the same technique as above, we can show that a positive real number x1=n exists for all
n 2 N and all x > 0. That is, for each x > 0, there exists a positive real number r such that rn = x.
The proof is left as an exercise.
1.2. THE SET OF REAL NUMBERS 27
1.2.2 Archimedean property
As we have seen, in any interval, there are plenty of real numbers. But there are also infinitely many
rational numbers in any interval. The following is one of the most fundamental facts about the real
numbers. The two parts of the next theorem are actually equivalent, even though it may not seem
like that at first sight.
Theorem 1.2.4.
(i) (Archimedean property) If x;y 2 R and x > 0, then there exists an n 2 N such that
nx > y:
(ii) (Q is dense in R) If x;y 2 R and x < y, then there exists an r 2 Q such that x < r < y.
Proof. Let us prove (i). We can divide through by x and then what (i) says is that for any real
number t := y=x, we can find natural number n such that n >t. In other words, (i) says that N  R is
unbounded. Suppose for contradiction that N is bounded. Let b := supN. The number b􀀀1 cannot
possibly be an upper bound for N as it is strictly less than b. Thus there exists an m 2 N such that
m > b􀀀1. We can add one to obtain m+1 > b, which contradicts b being an upper bound.
Now let us tackle (ii). First assume that x  0. Note that y􀀀x > 0. By (i), there exists an n 2 N
such that
n(y􀀀x) > 1:
Also by (i) the set A := fk 2 N j k > nxg is nonempty. By the well ordering property of N, A has a
least element m. As m 2 A, then m > nx. As m is the least element of A, m􀀀1 =2 A. If m > 1, then
m􀀀1 2 N, but m􀀀1 =2 A and so m􀀀1  nx. If m = 1, then m􀀀1 = 0, and m􀀀1  nx still holds
as x  0. In other words,
m􀀀1  nx < m:
We divide through by n to get x < m=n. On the other hand from n(y􀀀x) > 1 we obtain ny > 1+nx.
As nx  m􀀀1 we get that 1+nx  m and hence ny > m and therefore y > m=n.
Now assume that x<0. If y>0, then we can just take r =0. If y<0, then note that 0<􀀀y<􀀀x
and find a rational q such that 􀀀y < q < 􀀀x. Then take r = 􀀀q.
Let us state and prove a simple but useful corollary of the Archimedean property. Other
corollaries are easy consequences and we leave them as exercises.
Corollary 1.2.5. inff1=n j n 2 Ng = 0.
Proof. Let A := f1=n j n 2 Ng. Obviously A is not empty. Furthermore, 1=n > 0 and so 0 is a lower
bound, so b := inf A exists. As 0 is a lower bound, then b  0. If b > 0. By Archimedean property
there exists an n such that nb > 1, or in other words b > 1=n. However 1=n 2 A contradicting the fact
that b is a lower bound. Hence b = 0.

HOW BUSINESS AND MARKETING ARE CHANGING

We can say with some confidence that “the marketplace isn’t what it used to be.” It is
changing radically as a result of major forces such as technological advances, globalization,
and deregulation. These forces have created new behaviors and challenges:
Customers increasingly expect higher quality and service and some customization.
They perceive fewer real product differences and show less brand loyalty. They can
obtain extensive product information from the Internet and other sources, permitting
them to shop more intelligently. They are showing greater price sensitivity in their
search for value.
Brand manufacturers are facing intense competition from domestic and foreign
brands, which is resulting in rising promotion costs and shrinking profit margins.
They are being further buffeted by powerful retailers who command limited shelf
space and are putting out their own store brands in competition with national brands.
Store-based retailers are suffering from an oversaturation of retailing. Small retailers
are succumbing to the growing power of giant retailers and “category killers.”
Store-based retailers are facing growing competition from direct-mail firms; newspaper,
magazine, and TV direct-to-customer ads; home shopping TV; and the Internet.
As a result, they are experiencing shrinking margins. In response, entrepreneurial
retailers are building entertainment into stores with coffee bars, lectures, demonstrations,
and performances, marketing an “experience” rather than a product
assortment.
Company Responses and Adjustments
Given these changes, companies are doing a lot of soul-searching, and many highly
respected firms are adjusting in a number of ways. Here are some current trends:
➤ Reengineering: From focusing on functional departments to reorganizing by key
processes, each managed by multidiscipline teams.
➤ Outsourcing: From making everything inside the company to buying more products
from outside if they can be obtained cheaper and better. Virtual companies outsource
everything, so they own very few assets and, therefore, earn extraordinary rates of
return.
➤ E-commerce: From attracting customers to stores and having salespeople call on
offices to making virtually all products available on the Internet. Business-tobusiness
purchasing is growing fast on the Internet, and personal selling can
increasingly be conducted electronically.
➤ Benchmarking: From relying on self-improvement to studying world-class performers
and adopting best practices.
➤ Alliances: From trying to win alone to forming networks of partner firms.24
➤ Partner–suppliers: From using many suppliers to using fewer but more reliable
suppliers who work closely in a “partnership” relationship with the company.
➤ Market-centered: From organizing by products to organizing by market segment.
➤ Global and local: From being local to being both global and local.
➤ Decentralized: From being managed from the top to encouraging more initiative and
“intrepreneurship” at the local level.
Marketer Responses and Adjustments
As the environment changes and companies adjust, marketers also are rethinking
their philosophies, concepts, and tools. Here are the major marketing themes at the
start of the new millennium:
➤ Relationship marketing: From focusing on transactions to building long-term,
profitable customer relationships. Companies focus on their most profitable
customers, products, and channels.
16 CHAPTER1 MARKETING IN THE TWENTY-FIRST CENTURY
➤ Customer lifetime value: From making a profit on each sale to making profits by
managing customer lifetime value. Some companies offer to deliver a constantly
needed product on a regular basis at a lower price per unit because they will enjoy
the customer’s business for a longer period.
➤ Customer share: From a focus on gaining market share to a focus on building customer
share. Companies build customer share by offering a larger variety of goods to their
existing customers and by training employees in cross-selling and up-selling.
➤ Target marketing: From selling to everyone to trying to be the best firm serving welldefined
target markets. Target marketing is being facilitated by the proliferation of
special-interest magazines, TV channels, and Internet newsgroups.
➤ Individualization: From selling the same offer in the same way to everyone in the
target market to individualizing and customizing messages and offerings.
➤ Customer database: From collecting sales data to building a data warehouse of
information about individual customers’ purchases, preferences, demographics,
and profitability. Companies can “data-mine” their proprietary databases to detect
different customer need clusters and make differentiated offerings to each cluster.
➤ Integrated marketing communications: From reliance on one communication tool such
as advertising to blending several tools to deliver a consistent brand image to
customers at every brand contact.
➤ Channels as partners: From thinking of intermediaries as customers to treating them
as partners in delivering value to final customers.
➤ Every employee a marketer: From thinking that marketing is done only by marketing,
sales, and customer support personnel to recognizing that every employee must be
customer-focused.
➤ Model-based decision making: From making decisions on intuition or slim data to
basing decisions on models and facts on how the marketplace works.
These major themes will be examined throughout this book to help marketers and companies
sail safely through the rough, but promising, waters ahead. Successful companies
will change their marketing as fast as their marketplaces and marketspaces change, so
they can build customer satisfaction, value, and retention,

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Monopoly PC GAME

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Masud Rana Bangla Book

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Ashcche Cyclone_2
Bagher Khacha
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Bishchakra
Bondi Rana
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Charidike Shotru_Part 1 & 2
DurantoEagle_Part1
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I Love You_ Man_Part 1_ 2 & 3
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Jimmi
Kalo Naksha
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Major Rahat_Part 1_ 2
Mission Tel Aviv
Orokkhito Joloshima
Rana Shabdhan!
Red Dragon
Swapner Bhalobasha 2
Swarnodwip_Part1
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Bangla Book of Jafar Iqbal

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Abonil by Md. Jafar Iqbal
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Bigyani_Sofdor_Alir_Moha_Moha_Abishkar_by_Md_Jafar_Iqbal
Doler Naam Black Dragon by Md. Jafar Iqbal
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War Chess PC GAME

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Bejeweled 2 Cracked PC GAME

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29  is a very popular CARD GAME. Now I am giving you to full version 29 Game. Enjoy the game…….. 

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