Prove that for all $n\in \mathbb{N}$, it holds that for all real numbers $a,b\geq 0$, that $(\frac{a+b}{2})^n \leq \frac{a^n+b^n}{2}$
How to prove by induction that $\frac{a^n+b^n}{2}\geq\left(\frac{a+b}{2}\right)^n$?