I've started learning a bit about simplicial homology, recently. On the way to proving topological invariance, the book ('Elements of Algebraic Topology', Munkres) uses subdivisions and simplicial approximation. However the following simple statement is taken for granted in a proof, and I haven't been able to rigorously prove it myself. Namely
If $K$ is any finite subdivision of $\Delta^n$, where $\Delta^n$ is the standard $n$-dimensional simplex, then the reduced homology $\tilde H_p(K)$ of $K$ vanishes identically.
This is used in the proof of the existence of the subdivision operator (Theorem 17.2, on page 96 of my edition). I think it might be relatively easy to see... but I'm at a loss, nevertheless.
At this point in the book, the reduced homology of a simplex and its boundary have been computed. So the above is known to be true for $K=\Delta^n$. More generally, it has been proven that a cone always has trivial homology (if that's of any help). No other homology theory (like singular homology) has been introduced, yet.
I'd very much appreciate your help! Thank you.