Prove that $|x| + |y| \le |x-y| + |x+y| $

Question

I tried working from both sides:

$| |x| + |y|| = | |x| + |x| - |x| + |y|| = | 2|x| + |y| - |x|| \le |2x| + | |y| - |x| |$

$\Longrightarrow||x| + |y|| = |x| + |y| \le |2x| + | |y| - |x| |$

Then,

$|2x| = |x - y + x + y| \le |x-y| + |x+y| $

Can someone help me from this point on?

Answer 1

We have that by tringle inequality

$$2|x|=|x+y+x-y|\le |x+y|+|x-y|$$

$$2|y|=|x+y-x+y|\le |x+y|+|x-y|$$

then by addition

$$2|x|+2|y|\le 2|x+y|+2|x-y|$$