Hello all I am just wondering if my thoughts on this are correct.
Suppose we are just talking about some bounded set $$A \subseteq \mathbb{R}$$
we say $w$ is a sup of A iff it is an upper bound of A and $\forall \epsilon \gt 0$ there $\exists$ an $a_{\epsilon} \in A$ such that $$w-\epsilon \lt a_{\epsilon}$$
Then could we also say the same but instead using $\frac{\epsilon}{2}$
ie it is the supremum iff it is an upper bound and $\forall \frac{\epsilon}{2} \gt 0$ there $\exists$ an $a_{\frac{\epsilon}{2}}$ $\in A$ such that $$w-\frac{\epsilon}{2} \lt a_{\frac{\epsilon}{2}}$$