Proof
We consider two cases: if \(n\) is even or if \(n\) is odd.
Case 1: \(n\) is even. Then \(n = 2k\) for some integer \(k\text{.}\) This give
\begin{align*}
n^3 - n \amp = 8k^3 - 2k\\
\amp = 2(4k^2 - k)\text{,}
\end{align*}
and since \(4k^2 - k\) is an integer, this says that \(n^3-n\) is even.
Case 2: \(n\) is odd. Then \(n = 2k+1\) for some integer \(k\text{.}\) This gives
\begin{align*}
n^3 - n \amp = (2k+1)^3 - (2k+1)\\
\amp = 8k^3 + 6k^2 + 6k + 1 - 2k - 1\\
\amp = 2(4k^3 + 3k^2 + 2k)\text{,}
\end{align*}
and since \(4k^3 + 3k^2 + 2k\) is an integer, we see that \(n^3 - n\) is even again.
Since \(n^3 - n\) is even in both exhaustive cases, we see that \(n^3 - n\) is indeed always even.