I'm reading Milnor's Morse Theory and stumble upon a proof of a theorem below. My question in the end is only about topology.
$\textit{Theorem 3.5. }$ If $f$ is a differentiable function on a manifold $M$ with no degenerate critical points, and if each $M^a:=f^{-1}(-\infty, a]$ is compact, then $M$ has the homotopy type of a CW-complex, with one cell of dimension $\lambda$ for each critical points of index $\lambda$.
I try to summarize the part of the proof where I didn't understand as follows : with $M^{a_i}:=f^{-1}(-\infty, a_i]$, suppose that we have infinite sequences of homotopy equivalences $M^{a_i} \to K_i$ (where $K_i$ are CW-complexes),
each extending the previous one. The claim is that if $K=\bigcup_{i=1}^{\infty} K_i$ equipped with direct limit topology, and $g: M \to K$ be the limit map, then $g$ induces isomorphisms of homotopy groups of all dimensions.
My Question: I have a very limited knowledge of Algebraic Topology so i can't understand how such claim is true. I don't even know what is "limit map" and how to working with direct limit. I really appreciate if somebody can give me some reference about this (so i can learn it by myself) and maybe a short explanation if possible on why $g$ induces such isomorphisms. I've found a similar question of this issue here but I still couldn't get accessible explanation. Thank you.
For those who want to see the full proof in the text, here it is.
