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. Andsince S is nonempty, there existss S. By definition, we have inf S sand 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..

1. 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.
Andsince S is nonempty, there existss S.
By definition, we have inf S sand 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.