Homology of product of topological space and sphere is direct sum of homologies. 
Show that for $i > n \in\mathbb{N}$: $$H_{i}\left(X \times \mathbb{S}^{n}\right) \simeq H_{i}\left(X\right) \oplus H_{i - n}\left(X\right).$$

My first idea motivated by $n=0$ case (which is obvious) was to try induction but I cannot see how to perform next step. However question seems to be very neat so I decided to share it.

The result follows directly from the Kuenneth formula, since $H_p(S^n) = {\mathbb{Z}}$ for $p = 0, n$ and vanishes in all other dimensions.
