# How to prove a number is a supremum of a set

1. Oct 4, 2009

### gotmilk04

1. The problem statement, all variables and given/known data
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

2. Relevant equations

3. 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.

2. Oct 4, 2009

### jbunniii

Re: suprenums

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?