Properties of Supremum

Question

Theorem $5$. If $M$ is $\sup S$ and $\epsilon > 0$ , then there is at least one number $s$ in $S$ such that $M −\epsilon < s \le M$

I know theorems are always true... but if $\sup S$ is a "unique" number which is the largest number in the upperbound of $S$, how can $s \leq M$? Shouldn't it be $s < M$

Answer 1

It is possible for the supremum to be in the set.

For example let $S$ be $ \{2\}$.

Then the supremum is $2$ and we have to pick $s=M$.

Remark:

Supremum is the $\color{blue}{\text{least}}$ upper bound.