For $X \sim N(\mu, \sigma^2)$ the expected value of $|X|$ is calculated as
$$E|X| = \sigma \sqrt{\frac{2}{\pi}} e ^{-\mu^2/2\sigma^2} + \mu\left(1 - 2\Phi\left(-\frac{\mu}{\sigma}\right)\right)$$
I would like to see a proof that says for $\sigma$ fixed and $\mu \in (-\infty, \infty)$
$$\sigma\sqrt{\frac{2}{\pi}} \leq E|X|$$
with respect to $\mu$. A citation would also be great to see! Thanks for the help!