Definitions of supremum

This question is for me to better understand the beginning of a real analysis course.

We are provided with two definitions of supremum as follows:
Def 1 : Let $S$ be a set in $\mathbb{R}$ be bounded above, then $m$ is called the least upper bound (supremum) if $m \ge s\space, \forall s\in S $ and if $m'$ is some other upper bound, then $m < m'$
Def 2: Let $S$ be a set in $\mathbb{R}$ be bounded above, then $m$ is a supremum if for some arbitrary $\epsilon>0$ $\exists s \in S, m-\epsilon < s$
I understand how both statements are true, however, would it be possible to prove the Def 2 based on Def 1?

Any hint is appreciated!

Solution 1:

Suppose $m$ is a supremum by definition 1. Suppose $\exists$ $\epsilon>0$ such that $m-\epsilon\geq s$ $\forall s\in S$, then $m'=m-\epsilon$ is another upper bound so it must be $m<m'=m-\epsilon$ which is impossible. So we must have $\forall \epsilon>0$, $m-\epsilon <s$ $\exists s\in S$.

This proves definition 2 in terms of definition 1.