<s>[INST] Proof $|x+y|+|x-y| \geq |x| + |y|$ Proof that for all $x,y \in \mathbb{R}$:
$$|x+y|+|x-y| \geq |x| + |y|$$
My (failed) attempt:
$$|x + y| \leq |x| + |y|$$
$$|x - y| \leq |x| + |y|$$
$$|x + y|+|x - y| \leq 2(|x| + |y|)$$
Thank you
 [/INST] We have,
$$\color{red}{(|x+y|+|x-y|)^2 = 2(x^2+y^2) +2|x^2-y^2|}$$ Whence, 
\begin{align}&2|x^2-y^2| \geq -x^2-y^2+2|x| |y|= -(|x|-|y|)^2\\
&\Longleftrightarrow(x^2+y^2) +2|x^2-y^2|\geq 2|x||y|\\
&\Longleftrightarrow 2(x^2+y^2) +2|x^2-y^2|\geq x^2+y^2+2|x||y|\\
&\Longleftrightarrow (|x+y|+|x-y|)^2 \geq (|x| + |y|)^2\\
&\Longleftrightarrow|x+y|+|x-y| \geq |x| + |y|
 \end{align}
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