 Holstein-Primakoff matrix-element proof

On $0\le n\le 2S$, the raising/lowering amplitudes are $\sqrt{n(2S-n+1)}$ and $\sqrt{(n+1)(2S-n)}$. Their squared difference gives $[S^{+},S^{-}]=2(S-n)$. The order of the number-dependent square root is essential.