Let S be a nonempty bounded subset of R i Prove inf S sup S

Let S be a non-empty bounded subset of R.

(i) Prove inf S sup S.

(ii) What can you say about S if inf S = sup S.

Solution

i) Since S is bounded and nonempty, both inf S and sup S exist.

And since S is nonempty, there exists s S.

By definition, we have inf S s and s sup S.

Thus, from above two inequalities, inf S s sup S.

ii) From above, inf S s sup S

and if inf S = sup S

This means that we must have equality in each case; i.e.,

inf S = s = sup S.

Since inf S is a lower bound for S, there can be no elements in S less than s.

Also since sup S is an upper bound for S, there can be no elements in S greater than s.

Thus S has exactly one element - s itself.