Recall the homotopy excision theorem, as stated in Hatcher (Theorem 4.23): Let $X$ be a CW complex decomposed as the union of subcomplexes $A$ and $B$ with nonempty connected intersection $C = A \cap B$. If $(A,C)$ is $m$-connected and $(B,C)$ is $n$-connected for some $n,m \geq 0$, then the map $\pi\_i(A,C) \rightarrow \pi\_i(X,B)$ is an isomorphism for $i<m+n$ and a surjection for $i=m+n$.


The proof of this theorem in Hatcher is fairly complicated (but elementary). The other sources I've looked at (e.g. May's book) give similar proofs. Are there other proofs? As an example, the first application Hatcher gives is to prove the Freudenthal suspension theorem, and my favorite proof of this uses the Serre spectral sequence. More generally, I often find proofs of basic homotopy theoretic results clearer if they use the Serre spectral sequence or other such things rather than being overly "elementary". But I'd also be interested in alternate elementary proofs.


The only other proof I know of is Rezk's proof using homotopy type theory, but I can't make heads or tails of it.


There is an earlier MO question [here](https://mathoverflow.net/questions/262207/are-there-intuitively-clear-and-not-technical-proofs-of-homotopy-excision-theore) that is sort of along the same lines, but it includes desiderata like the proof being "ideologically profound" that I certainly am not looking for (in fact, I don't even know what this means, to be honest).



Alternate proofs of homotopy excision theorem
The proof of Theorem 9.3.5 (especially the part on page 486) in Spanier's "Algebraic Topology" may be more to your liking. It presumes you have already established the relative Hurewicz theorem, e.g. by Serre spectral sequence methods.


