Proof

If $ E_\infty $ is finite, then $ P_\infty $ is always zero

$ P_\infty\equiv\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^T|x(t)|^2dt $

$ P_\infty\equiv(\lim_{T\to\infty}\frac{1}{2T})*(\lim_{T\to\infty}\int_{-T}^T|x(t)|^2dt) $

Because $ E_\infty\equiv\lim_{T\to\infty}\int_{-T}^T|x(t)|^2dt $, it follows that by substitution

$ P_\infty=(\lim_{T\to\infty}\frac{1}{2T})*E_\infty $

$ P_\infty=(\lim_{T\to\infty}\frac{1}{2T})*(\lim_{T\to\infty}E_\infty) $

$ P_\infty=\lim_{T\to\infty}\frac{E_\infty}{2T} $

This limit will always evaluate to zero as long as $ E_\infty $ is finite.

$ \therefore $ If $ E_\infty $ is finite, then $ P_\infty $ is always zero $ \square $

-Adam Siembida