<s>[INST] Which is larger, $\sqrt{3} + \sqrt{5}$ or $\sqrt{2} + \sqrt{6}$? The clue given by the text is to "use the fact that $\sqrt{x}$ is increasing."
I was able to get the correct answer here by squaring both expressions. But I don't think I made use of the text-prided clue, so am wondering if perhaps I'm missing the "deeper" lesson. 
What do you think the author is hinting at?
Thanks!
 [/INST] Since $x\mapsto\sqrt{x}$ is an increasing function we have the following
\begin{align*}
2\sqrt{15}&>2\sqrt{12}\\
8+2\sqrt{15}&>8+2\sqrt{12}\\
3+2\sqrt{15}+5&>2+2\sqrt{12}+6\\
(\sqrt{3}+\sqrt{5})^2&>(\sqrt{2}+\sqrt{6})^2
\end{align*}
Therefore, $\sqrt{3}+\sqrt{5}>\sqrt{2}+\sqrt{6}$.
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