Arithmetic mean is greater than or equal to the geometric mean

maths

Statement

Lemma

For $w_1, \ldots, w_k \geq 0$ we have the arithmetic mean is greater than or equal to the geometric mean. In otherwords

$$\frac{1}{k} \sum_{i=1}^k w_i \geq \left ( \prod_{i=1}^k w_i \right )^{1/k}.$$

Proof

All the proofs for this are really horrible ….