I have a question concerning the proof of theorem 3.5 in Milnor's Morse Theory.
This theorem states that if $f$ is a differentiable function on a Manifold M with no critical points, and if each $M^a = \{x\in M | f(x)\leq a \} $ is compact then $M$ has the homotopy type of a CW-complex with one cell of dimension $\lambda$ for each critical point of index $\lambda$
Basically the proof shows via induction and appealing to previous theorems that $M^a$ has the homotopy type of a CW complex satisfying the conclusion above. But then Milnor says "If $M$ is compact this completes the proof".
I don't understand how this just follows directly? Is there some theorem involving Compact manifolds or CW complexes that I am missing? Does $M^a$ have to be homotopic to $M$ for some $a$ if $M$ is compact?
If $M$ is compact, one of the $M^a$ is equal to $M$!