# Morse lemma implies nondegenerate critical points are isolated

It might be a stupid question. I don't quite know how we get the conclusion " Non-degenerate critical points are isolated" from Morse lemma. I know around each non-degenerate critical point we have a neighborhood of it with a local coordinate system satisfying the Morse lemma but how we know two local coordinates cannot intersect? I don't see where is the contradiction that a non-degenerate critical point cannot be an accumulating point? Thanks for any help!

If $p$ is a nondegenerate critical point with $f(p) = c$ then in some neighborhood $U$ of $p$ with coordinates $u_1, \ldots, u_{n}$ we can write $f = c - u_{1}^{2} - \cdots - u_{\lambda}^{2} + u_{\lambda + 1}^{2} + \cdots + u_{n}^{2}$ where $\lambda$ is the index of $p$. In these coordinates we will have $\frac{\partial f}{\partial u_i} = \pm 2 u_i$, so we see that $\frac{\partial f}{\partial u_i} = 0$ for all $i$ only when $u_i = 0$ for all $i$, which occurs only at $p$. Hence the only critical point of $f$ within $U$ is $p$.