We define the even and odd parts of a sequence \(x(n), \mathcal{E}\{x(n)\}\) and \(\mathcal{O}\{x(n)\}\) respectively, as

\[\begin{aligned}\mathcal{E}\{x(n)\} & =\frac{x(n)+x(-n)}{2} \\\mathcal{O}\{x(n)\} & =\frac{x(n)-x(-n)}{2}\end{aligned}\]

Show that

\[\sum_{n=-\infty}^{\infty} x^{2}(n)=\sum_{n=-\infty}^{\infty} \mathcal{E}\{x(n)\}^{2}+\sum_{n=-\infty}^{\infty} \mathcal{O}\{x(n)\}^{2}\]