# Generalization of Morse Lemma that appears in 'singular points of complex hypersurfaces'

In the book 'singular points of complex hypersurfaces', lemma 2.12, Milnor claims the following version of the Morse Lemma:

$$Let 0\in M\subset \mathbb{R}^{m}$$ be a smooth manifold, and $$r:\mathbb{R}^{m}\to\mathbb{R}$$ be a smooth function such that 0 is a non-degenrate critical point of $$r$$ with index $$0$$. Then there exists a coordinate system $$u_1,...,u_m$$ for $$\mathbb{R}^{m}$$ next to zero such that the following holds:

1) $$M=\{u_{k+1}=\dots=u_{m}=0\}$$

2) $$r=u_1^2+\dots+u_m^2$$

And he leaves the proof for the reader. Unfortuantely I can't really prove this (of course it is easy to satisfy each requirment seperetaly). I have tried these things:

A) Take a coordinate system where $$M$$ is of the form desired in 1). Fix the last $$m-k$$ coordinates. Then one can choose the first $$k$$ coordinates such that $$r(\cdot,u_{k+1},\dots,u_{m})=u_1^2+\dots +u_k^2$$. I can't move forward from this.

B)Looking at the proof of the morse lemma, but it is rather deterministic and I don't see how it can be sharpened to get this result.

C) Searching for a proof online, but no luck.

Any answer, hint or reference will be appreciated. Thank you!