A proof that could be spelled out more

[stratisMarkou]

cam-notes/IB_E/metric_and_topological_spaces.tex

Line 1683 in 0c1046b

Since $X$ is compact, $\im f$ is compact. Let $\alpha = \max \{\im f\}$. Then $\alpha \in \im f$. So $\exists x\in X$ with $f(x) = \alpha$. Then by definition $f(x) \geq f(y)$ for all $y\in X$.

I might be missing something, but to me it looks like the proof of this result is not complete / fully spelled out. It uses the fact that max(im f) exists, which is true because im f is a closed bounded subset of the reals, but none of these properties are pointed out here.