Lemma

For any sequence of $k$ nonnegative real numbers $b_{1},\dots,b_{k}$ $$\left( \prod_{i=1}^{k}b_{i} \right)^{\frac{1}{k}} \le \frac{1}{k}\sum_{i=1}^{k}b_{i}$$where equality holds if and only if $b_{i}=b, i=1,\dots,k$.