For a set of non-negative numbers, the [[arithmetic mean]] will never be exceeded by the [[geometric mean]] $\quad \sqrt{x y} \leq \frac{x+y}{2}$ #### Proof ##### Using [[Cauchy-Schwarz-Buniakowsky Inequality]] Take the [[dot product]] of $\boldsymbol v = (\sqrt x,\sqrt y)$ and $\boldsymbol w = (\sqrt y,\sqrt x)$ $\boldsymbol v \cdot \boldsymbol w=\sqrt{xy}+\sqrt{xy}=2\sqrt{xy} $ and know (by [[Cauchy-Schwarz-Buniakowsky Inequality]]) that it must be less than the product of the [[vector length|magnitudes]] of $\boldsymbol v$ and $\boldsymbol w$, both of which are $\sqrt{x+y}$ $\|\boldsymbol v\|=\sqrt{x+y} \qquad \|\boldsymbol w\|=\sqrt{x+y} $ $\|\boldsymbol v\|\|\boldsymbol w\|=x+y$ $\begin{aligned} |\boldsymbol{v} \cdot \boldsymbol{w}| &\leq\|\boldsymbol v\|\|\boldsymbol w\|\\ 2\sqrt{xy}&\leq x+y\\ \sqrt{xy}&\leq\frac{x+y}{2} \end{aligned}$