1. For each subset of , give its supremum and its maximum, if they exist. Otherwise, write “none.” 2. Let S be a nonempty bounded subset of  and let m = sup S. Prove that m∈S iff m = max S. 3. (a) Let...

1. For each subset of
, give its supremum and its maximum, if they exist. Otherwise, write “none.”

2. Let S be a nonempty bounded subset of
 and let m = sup S. Prove that m∈S iff m = max S.

3. (a) Let S be a nonempty bounded subset of
. Prove that sup S is unique.

(b) Suppose that m and n are both maxima of a set S. Prove that m = n.