I need some help with a statement from Milnor's Morse theory book!
While studying the proof of a theorem I got stuck.
That's what we know: Let $f$ be a differentiable function on a manifold $M$ with no degenerate critical points, and so that for all $a \in \mathbb{R}$, $M^{a}=\{p\in M: f(p)\leq a\}$ is compact.
The author says: " Let $c_{1}<c_{2}<c_{3}<...$ be the critical values of $f:M \rightarrow \mathbb{R}$. The sequence $\{c_{i}\}$ has no cluster point since each $M^{a}$ is compact. The set $M^{a}$ is vacuous for $a<c_{1}...$"
I proved that the set of critical values of $f$ is countable and that exists a minimum. Then I showed that the mentioned sequence has non cluster point but I did not manage in showing that he set "$M^{a}$ is vacuous for $a<c_{1}...$". What I used to show the first two statement are essentially: the fact that non degenerate critical points for a smooth real valued function on a manifold are isolated(consequence of Morse's lemma), the regularity of the function $f$ and the compactness of $M$.
I thought of showing the last observation by a "proof by contradiction" supposing it exists an $a\in \mathbb{R}$ for which $M^{a}$ is not vacuous and $a<c_{1}$ and then finding a critical point for $f$: $p\in M^{a}$, leading to a contradiction. I did not really get so far in that...Do you have any hints? Suggestions?
Thanks