<s>[INST] Why does the triangle inequality seem to be false when rewritten based on $|x| = \sqrt{x^{2}}$? The triangle inequality is $$|x + y| \leq |x| + |y|.$$
Also, we know that $|x| = \sqrt{x^{2}}$. Then,
\begin{align*}
    \sqrt{x + y} &\leq \sqrt{x} + \sqrt{y} \\
    x + y &\leq x + y + 2\sqrt{xy} \\
    0 &\leq 2\sqrt{xy} \\\\
    \sqrt{xy} &\geq 0
\end{align*}
Now, we can see that the inequality $|x + y| \le |x| + |y|$ holds for real $x$ and $y$, but $\sqrt{xy} \geq 0$ does not for $x < 0, y > 0$ or $x > 0, y < 0$.
What seems to be the problem? Is it the statement $|x| = \sqrt{x^{2}}$ or is it much more than that?
 [/INST] The statement you have written using $|x|=\sqrt {x^2}$ is wrong. The correct statement would be: $$\sqrt {(x+y)^2}\leq \sqrt {x^2}+\sqrt {y^2}$$ which, indeed, does hold for all real $x,y$, as can be verified by squaring and simplifying.
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