\(-|x| \leq x \leq |x|\) and \(-|y| \leq y \leq |y|\) \(\implies\) \(-|x|-|y| \leq x+y \leq |x|+|y|\) \(\implies\) \(|x+y| \leq |x| +|y|\) for any real \(x,y\)

The last implication comes from the fact: \(|x| \leq a \leftrightarrow -a \leq x \leq a\) for some \(a \geq 0\)

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