Proof of being a compact set

Question

I'm trying to solve this problem but I'm really stuck and it would be nice if someone can explain me proof or any hint for this problem.

Let $X \subset\mathbb R^N$ be a nonempty compact set, and $f: X \rightarrow\mathbb R$ a continuous function. Show that

$\operatorname{argmax}\limits_{x\in X}f(x) = \{ x \in X \mid f(x) \ge f(y) \text{ for all } y \in X\}$ is a compact set

Answer 1

Since $X$ is compact, the image in $\mathbf{R}$ is compact (as $f$ is continuous). Let $y$ be the maximum of the image of $f$.

Then $f^{-1}(y)$ is a closed set in a compact space, hence compact.