Periodicity of transfer function of FIR filter proof (Parks and Burrus, Digital Filter Design)
$$\sum_{n=0}^{N-1}h(n)e^{-j\omega n}e^{-j2\pi n}=\sum_{n=0}^{N-1}h(n)e^{-j\omega n}\cdot1$$

Since
\begin{align}
e^{-j2\pi n}&amp;=\cos(-2\pi n) + j\sin(-2\pi n)\\
&amp;=\cos(2\pi n) - j\sin(2\pi n)\\
&amp;=1-0\\
&amp;=1
\end{align}