I saw a claim in an old version of the book of Fuchs and Fomenko saying that every CW complex is homotopically equivalent to a simplicial complex.
I would like to see a proof of this claim. Hopefully not the one on Hatcher's book (Th 2C.5, page 182) nor on Gray's book.
Essentially the way I thought about proving this statement is by showing that the statement is trivial on the null and one dimensional skeletons, and then say that the closure of every cell of dimension at least 2 admits a triangulation, and every $n$-dimensional triangle is homeomorphic to the $n$-simplex, making the two underlying topological spaces homeomorphic and therefore homotopy equivalent.
My issue is that this argument seems too short compared to (Hatcher, Th 2C.5, page 182), and proves something that is stronger than the actual claim, that is, I prove that every CW complex is homeomorphic to a simplicial one. Whereas this does sound plausible for a pleb like me, I think this would potentially make a lot of theory redundant, for example we wouldn't need both a simplicial and a cellular approximation theorem, which makes me think that I am definitely missing something important.
Thanks in advance!