I'm trying to prove Reeb's theorem as stated in Milnor's Morse Theory. That is, suppose we have an $n$-manifold $M$ together with a smooth function $f$ with exactly two critical points (both non-degenerate), then $M$ is homeomorphic to $S^n$.
I understand the most crucial part. That is, let $f$ be normalized such that the critical points are mapped to $0$ and $1$ respectively. We can then prove that $f^{-1}[0,a]$ is diffeomorphic to $f^{-1}[0,b]$ for any $0<a,b<1$ by constructing a certain vector field out of the gradient of $f$ and then using the flow of this vector field as the required diffeomorphism. Then if $f^{-1}[0,\epsilon]$ and $f^{-1}[1-\epsilon,1]$ are homeomorphic to a disk, we have that $M$ must be the union of two disks glued along their boundary, making $M$ homeomorphic to $S^n$.
Now my problem is the part where there is an $\epsilon>0$ such that $f^{-1}[0,\epsilon]$ is homeomorphic to an ($n$-dimensional) disk. Milnor uses a lemma (Morse Lemma) which constructs quadratic coordinates in some open neigborhood of $f^{-1}(0)$, but this seems to far too strong. So I was wondering if it's possible to prove this without resorting to that lemma.
My idea is first of all to note that $f^{-1}[0,\epsilon]$ is contractible since we can simply consider the family $f^{-1}[0,t]$ for $t\in[0,\epsilon]$. So if we shrink $f^{-1}[0,\epsilon]$ such that it fits inside a single chart, we can consider it as an closed contractible subset of $\mathbb{R}^n$. However this is not enough to make it homeomorphic to a disk.
Another approach would be to prove that $f^{-1}[\epsilon,a]$ is homeomorphic to $S^{n-1}\times [0,1]$, and then argue that since for any $0<t<\epsilon$, $f^{-1}[t,a]$is a deformation retract of $f^{-1}[\epsilon,a]$, so $f^{-1}[0,a]$ would be homeomorphic to $S^n\times [0,1]$ with $S^n\times\{0\}$ identified to a point. This space is then homeomorphic to a disk as well.
Milnor also remarks that the non-degeneracy condition on $f$ is not essential. The way I would like to prove it doesn't use non-degeneracy either. He refers to two sources that supposedly give a proof that don't use the non-degeneracy condition, but I wasn't able to find a copy online and my university's library doesn't stock any copies.
Any thoughts?
Edit:
In the end I was able to find a book after all. In Saaty's Lectures on Modern Mathematics vol. 2 Milnor has written a chapter on differential topology. In this section he gives a proof that doesn't require the non-degeneracy condition.
He uses a lemma by Brown and Stalling that says that for a paracompact manifold $M$, if any compact subset is contained in an open set diffeomorphic to $\mathbb{R}^n$, then $M$ is itself diffeomorphic to $\mathbb{R}^n$.
Let $x_0$, $x_1$ be the critical points, and let $U$ be a neigborhood of $x_0$ diffeomorphic to $\mathbb{R}^n$. Then using the gradient we can always stretch $U$ such that it covers any compact subset of $M-x_1$. Hence $M-x_1$ is diffeomorphic to $\mathbb{R}^n$. Now $M$ is homeomorphic to the one point compactification of $\mathbb{R}^n$ which is in turn homeomorphic to $S^n$, completing the proof.