A Morse function on a compact manifold has finitely many critical points

Question

We still have a problem with the Morse lemma.

Let $u$ be a non-degenerate critical point of the function $f : \mathbb{M} \to \mathbb{R}.$ There are local coordinate with $u = (0, \dots, 0)$ such that $$ f(x) = f(u) - x_1^2 - \dots - x_i^2 + x_{i+1}^2 + \dots + x_n^2 $$ for every point $x = (x_1, \dots, x_n)$ in a small neighborhood of $u$.

A consequence of the Morse lemma is that non-degenerate critical points are isolated. In particular, a Morse function on a compact manifold has finitely many critical points.

The first part of the consequence we could understand but the second one "A Morse function on a compact manifold has finitely many critical points" we coundn't get it.

Could you please give us a hint? Thank you!

Answer 1

HINT:

You should prove that a closed subset consisting of isolated points of a compact space is finite. First, find an open cover of the space such that each open set of that cover intersects the set in at most one point. Now, use compactness

Answer 2

Suppose the set of critical points $S$ is infinite, then by the Bolzano-Weierstrass property, every infinite subset of a compact space has a limit point. Let $p$ be the limit point of $S$. If $p\in S$, we contradict that $p$ is isolated; if $p\notin S$, there are two possibilities. If $df_p\neq 0$, we contradict the continuity of the partial derivatives of $f$. If $\det Hess(f)_p=0$, we contradict the continuity of the second-order partial derivatives of $f$.