TITLE: Outline of the proof that Cech cohomology and singular cohomology coincide on any locally contractible space
QUESTION [8 upvotes]: If $X$ is paracompact and locally contractible, then singular cohomology and Cech cohomology of $X$ coincide, with coefficients in any abelian group.

I hear that this is a classical result but I fail to see a clear proof in the literature. Dan Petersen says here that it is in Spanier's book. (I am assuming it is the Algebraic topology book by Spanier). I checked it twice roughly but somehow I could not find it. It would be a good idea to give some outline even if it is there.
Can someone give an outline of how this proof goes? There is some proof by using a double cochain complex here. Is there a proof that does not use this double cochain complex set up? Please do not assume any familiarity with spectral sequences.

REPLY [9 votes]: It is in Spanier's book, but you have to do two steps. On p. 334 he proves that Cech and Alexander-Spanier cohomology coincide, on p. 340 he proves that Alexander-Spanier and singular cohomology coincide.