Page 70 - DMTH401_REAL ANALYSIS

Real Analysis

Notes The collection G = {] – n, n[ : n N} is an open cover of R but does not admit of a finite subcover
of R. Therefore the set R is not a compact set.
Thus you have seen that every finite set is always compact. But an infinite set may or may not be
a compact set. The question, therefore, arises, "What is the criteria to determine when a given set
is compact?" This question has been settled by a beautiful theorem known as Heine-Borel Theorem
named in the honour of the German Mathematician H.E. Heine [1821-1881] and the French
Mathematician F.E.E. Borel [1871-1956], both of whom were pioneers in the development of
Mathematical Analysis.
We state this theorem without proof.
Theorem: Heine-Borel Theorem

Every closed and bounded subset of R is compact.
The immediate consequence of this theorem is that every bounded and closed interval is compact.

Self Assessment

Fill in the blanks:
1. A number p is said to be a ............................ of real numbers if every neighbourhood of p
contains at least one point of the set S different from p.
2. Let S be an infinite and bounded subset of R. Since A is bounded, therefore A has both a
lower bound as well as an .............................
3. A set is said to be closed if it contains all its .......................
4. A set is closed if and only if its .......................... is open.
5. Let S be a set and {G } be a collection of some open subsets of R such that ........................

Then {G} is called an open cover of S.
6. A set is said to be compact if every open cover of it admits of a ............................ of the set.

3.8 Summary

 We have defined the absolute value or the modulus of a real number and discussed certain
related properties. The modulus of real number x is defined as
|x| = x if x  0
= –x if x < 0.

Also, we have shown that
|x – a| < d  a – d < x < a + d
 We have discussed the fundamental notion of NBD of a point on the real line i.e. first we
have defined it as a – neighbourhood and then, in general, as a set containing, an open
interval with the point in it.
 With the help of NBD of a point we have defined, an open set in the sense that a set is open
if it is a NBD of each of its points.
 We have introduced the notion of the limit point of a set. A point p is said to be a limit
point of a set S if every NBD of p contains a point of S different from p. This is equivalent
to saying that a point p is a limit point of S if every NBD of p contains an infinite number
of the members of S. Also, we have discussed Bulzano-Weiresstrass theorem which gives

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