Understanding a math statement I want to understand the following statement:

Given a subset of real numbers $\mathcal{M}$, we find the number $C$ which is the largest $R>0$ such that $\forall \gamma>0$,  $ \exists m \in \mathcal{M}$ such that $m>R-\gamma$

Is this equivalent to $\mathcal{M}=(0,C]$, or $\mathcal{M}=(0,C)$?

Both and neither.  It is neither because we are not given that $\mathcal{M}$ is an interval.  It could be just a single point, it could be $\{1-\frac 1n|N \in \Bbb N\}$, it could be an interval plus some stray points, it could be any set.  To make the definition work it must be bounded above.  $C$ is then the least upper bound of $\mathcal{M}$.  If $\mathcal{M}$ has a maximum, $m$ is it and $m=C$.  If $\mathcal{M}$ approaches a limit at the top, $C$ is that limit, so $\mathcal{M}$ could be either open or closed at the top.
