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

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!