Page 36 - DMTH401_REAL ANALYSIS

Real Analysis

Notes You can easily verify that a subset of a bounded set is always bounded since the bounds of the
given set will become the bounds of the subset.
Now consider any two bounded sets say S = {1, 2, 5, 7} and T = {2, 3, 4, 6, 7, 8}. Their union and
intersection are given by
S  T = {1, 2, 3, 4, 5, 6, 7, 8}
and

S  T = {2, 7}.
Obviously S  T and S  T are both bounded sets. You can prove this assertion in general for any
two bounded sets.

Task Prove that the union and the intersection of any two bounded sets are bounded.

Now consider the set of negative integers namely
S = {–1, –3, –2, –4, ....}.
You know that –1 is an upper bound of S. Is it the only upper bound of S? Can you think of some
other upper bound of S? Yes, certainly, you can. What about 0? The number 0 is also an upper
bound of S. Rather, any real number greater than –1 is an upper bound of S. You can find
infinitely many upper bounds of S. However, you can not find an upper bound less than –1. Thus
–1 is the least upper bound of S.

It is quite obvious that if a set S is bounded above, then it has an infinite number of upper
bounds. Choose the least of these upper bounds. This is called the least upper bound of the set S
and is known as the Supremum of the set S. (The word ’Supremum’ is a Latin word). We formulate
the definition of the Supremum of a set in the following way:
Definition 6: The Supremum of a Set

Let S be a set bounded above. The least of all the upper bounds of S is called the least upper
bound or the Supremum of S. Thus, if a set S is bounded above, then a real number m is the
supremum of S if the following two conditions are satisfied:
(i) m is an upper bound of S,
(ii) if k is another upper bound of S, then m 5  k.

Task Give an example of an infinite set which is bounded below. Show that it has an
infinite number of lower bounds and hence develop the concept of the greatest lower
bound of the set.

The greatest lower bound, in Latin terminology, is called the Infimum of a set.
Let us now discuss a few examples of sets having the supremum and the infimum:

Example: Each of the intervals ]a, b[, [a, b] ]a, b], [a, b[ has both the supremum and the
infimum. The number a is the infimum and b is the supremum in each case. In case of [a, b] the
supremum and the infimum both belong to the set whereas this is not the case for the set ]a, b[.
In case of the set ]a, b], the infimum does not belong to it and the supremum belongs to it.
Similarly, the infimum belongs to [a, b[ but the supremum does not belong to it.

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