I am looking for a proof as elementary as possible of the fact that $H_k(X,\mathbb{Z})\bigotimes_{\mathbb{Z}} \mathbb{R}$ injects into $H_k(X,\mathbb{R})$, where $H_k$ denotes the homology group of a topological space (in my case, a compact manifold even). I know that this is a trivial consequence of the universal coefficient theorem for homology (i.e. 3A.3 in Hatcher), but I am wondering if there is a simpler proof for this specific case.
Thanks a lot!