I am trying to find the way to reduce the standard expression to compute the power of a generic sequence $x(n)$:
$$P_{\text{x}}= \lim\limits_{N \to \infty}\frac{1}{2N + 1}\sum\limits_{n=-N}^{N}|x(n)|^2\text,$$
when $x(n) = x(n + N_0)$ ($N_0$-periodic sequence), into the simplified formula
$$P_{\text{x}}= \frac{1}{N_0}\sum\limits_{n=0}^{N_0-1}|x(n)|^2\text.$$
How can I demonstrate that second one could be obtained from the first one, using the hypothesis of periodicity? I've tried something like this, but I get stuck in after $\text{(iii)}$:
$$\begin{align} N &= N_0\tag{i}\\ P_{\text{x}}&= \lim\limits_{N_0 \to \infty}\frac{1}{2N_0 + 1}\sum\limits_{n=-N_0}^{N_0}|x(n)|^2\\ &= \lim\limits_{N_0 \to \infty}\frac{1}{2N_0 + 1}\sum\limits_{n=-N_0}^{N_0}|x(n + N_0)|^2\tag{ii}\\ &m = n + N_0\text{, so }\\ P_{\text{x}}&= \lim\limits_{N_0 \to \infty}\frac{1}{2N_0 + 1}\sum_{m=0}^{2N_0}|x(m)|^2\tag{iii} \end{align} $$