Property of supremum for any real exponent

Question

For any $\alpha>0$ and $f:\Omega\to(0,\infty)$ bounded, is it true that $\sup f^{\alpha}\leq (\sup f)^\alpha$?

Also what about $\alpha<0$.

I know for $\alpha\in\mathbb{N}$, it holds.

But for any $\alpha>0$, unable to prove or give counter-example.

Please help me.