Page 44 - DMTH401_REAL ANALYSIS

Real Analysis

Notes You may follow the method of indicating by arrows for making a single list or you may follow
another path as indicated here. Accordingly, write down the elements of Q as they appear in the
+
figure by the arrows, while omitting those numbers which are already listed to avoid the
duplicates. We will have the following list:
+ { 1 1 1 2 3 }
Q = 1, 2 , 2, 3, 3 4 3 2 , 4, ..
,
,
,
=  A i (i = 1, 2, 3, ......),
i
which is countable by Theorem 3. Thus Q is countable.
+
Now let Q denote the set of all negative rational numbers. But Q and Q are equivalent; sets
– + –
because there is one-one correspondence between Q and Q , f: Q  Q , given by
+ – + –
f(x) = –x, " x  Q .
+
Therefore Q is also countable. Further {0} being a finite set is countable. Hence,
–
Q = Q  {0}  Q -
+
i
is a countable set. Thus, in fact, we have proved the following theorem:

Theorem 6: The set Q of all rational numbers is countable.
Proof: You may start thinking that perhaps every finite set is denumerable. This is not true. We
have not yet discussed the countability of the set of real numbers or of the set of irrational
numbers. To do so, we first discuss the countability of the set of real numbers in an interval with
end points 0 and 1, which may be closed or open or semi-closed.

Consider the real numbers in the interval ]0, 1[.
Each real number in ]0, l[ can be expressed in the decimal expansion. This expansion may be non-

terminating or may be terminating, e.g.
1
= .333, ......
3
is an example of non-terminating decimal expansion, whereas

1 1
= .25, = .5, ......,
4 2
are terminating decimal expansions. Even the terminating expansion can also be expressed as
non-terminating expansion in the sense that you can write
1
= .25 = .24999 .....
4
Thus, we agree to say that each real number (rational of irrational) in the ]0, 1[ can be expressed
as a non-terminating decimal expansion in terms of the digits from 0 to 9.
Suppose x ]0, 1[. Then it can be written as
x = .C C C .....
1 2 3
where c , c ,.... take their values from the set {0, l, 2, 3, 4, 5, 6, 7, 8, 9) of ten digits.
1 2
Similarly, let y be another, real number in (0, l). Then y can also be expressed as

y = .d d d .....
1 2 3

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