Prove $(a + b) \bmod n = (a \bmod n + b \bmod n) \bmod n$

Let $a = hn + (a \bmod n)$, $b = kn + (b \bmod n)$, $h,k\in \mathbb Z$. Then the left hand side

$$\begin{align*} (a+b)\bmod n =& [a+b-(h+k)n]\bmod n\\ =& [(hn +a\bmod n) + (kn+b\bmod n) - hn - kn]\bmod n\\ =& (a\bmod n + b\bmod n)\bmod n \end{align*}$$