<s>[INST] A quick way to prove the inequality $\frac{\sqrt x+\sqrt y}{2}\le \sqrt{\frac{x+y}{2}}$ Can anyone suggest a quick way to prove this inequality?
$$\frac{\sqrt x+\sqrt y}{2}\le \sqrt{\frac{x+y}{2}}$$
 [/INST] $2ab \leq a^2+b^2$ so $\frac{a+b}{2} \leq \sqrt{\frac{a^2+b^2}{2}}$. Now choose $a = \sqrt{x},b = \sqrt{y}$.
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