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!


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