How to prove a number is a supremum of a set

[gotmilk04]

Homework Statement

Prove that if
(i) $$\forall n$$$$\in N$$, u - (1/n) is not an upper bound of s
(ii) $$\forall n$$$$\in N$$, u + (1/n) is an upper bound of S
then, u = supS

Homework Equations

The Attempt at a Solution

It (i) and (ii) are true, then
$$\exists s$$$$\in S$$ s.t. u - (1/n) < s
and u+(1/n)>s for all s.
I'm not sure where to go from here.

 

[jbunniii]

How about a proof by contradiction?

Suppose $u > \sup S$. Then there exists $n \in \mathbb{N}$ such that $\sup S < u - 1/n < u$. Does this violate one of (i) or (ii)?

Next suppose $u < \sup S$. Can you rule this out as well?