<s>[INST] How is $\left|\frac{xy}{\sqrt{x^2+y^2}}\right| \leq \frac{\sqrt{|xy|}}{\sqrt{2}}$ $$\left|\frac{xy}{\sqrt{x^2+y^2}}\right| \leq \frac{\sqrt{|xy|}}{\sqrt{2}}$$
Does this apply in general, because here in this example i have $x \to 0, y \to 0$. Some inequality is used I believe to prove this one, but I do not see which.
 [/INST] $$\begin{align*}
(|x|-|y|)^2 &\ge 0\\
x^2 - 2|xy| + y^2 &\ge 0\\
x^2 + y^2 &\ge 2|xy|\\
\sqrt{x^2+y^2} &\ge \sqrt{2|xy|}\\
\frac{1}{\sqrt2} &\ge \frac{\sqrt{|xy|}}{\sqrt{x^2+y^2}}\\
\frac{\sqrt{|xy|}}{\sqrt2} &\ge \frac{|xy|}{\sqrt{x^2+y^2}}\\
\end{align*}$$
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