Theorem

Consider an arbitrary discrete-time memoryless additive-noise channel, whose output $Y_{i}$ at time $i$ is given by $$Y_{i}=X_{i}+Z_{i}$$where $\{X_{i}\}\perp\!\!\!\perp\{Z_{j}\}$ and $\{Z_{i}\}$ is iid, memoryless noise process, admitting pdf $f_{Z}$ on $\mathbb{R}$ with mean $0$ and variance $\sigma^{2}$. Let the channel be used with an input power constraint $P$. Then the channel capacity $C(P)$ satisfies: $$C(P)\ge C_{G}(P)= \frac{1}{2}\log_{2}\left(1+ \frac{P}{\sigma^{2}}\right)$$with equality iff the additive noise is Gaussian (i.e. $Z_{i}\sim\mathcal{N}(0,\sigma^{2}$)