Suprema and Infima of subsets of $\mathbb R$.

Question

Real Analysis Question (Suprema, Infima and $\mathbb R$)

Question:

Suppose $U$ is a non-empty subset of $\mathbb R$, bounded above, with supremum $s$. If $a$ is any number satisfying $a < s$, explain why there is some $u\in U$ with $a < u$.

Please can someone help with this question?

Answer 1

Hint: If the set $U$ had no element $>a$, then $a$ would be an upper bound for $U$.

Answer 2

If not, then $u \leq a$ for all $u\in U$, but this shows that $a$ is the sup, contridiction.