Question

Prove
|x+y| ≤ |x|+|y|

Answer 1

This is the triangle inequality, my friend.

Answer 2

I shall present a proof by exhaustion.

\[|x+y|\leq|x|+|y|\]Say that...\[{x\geq0}\land{y\geq0}\]So, we find...\[|x + y| =x + y= |x|+|y|\]Now, say we have...\[{x\lt0}\land{y\lt0}\]... so we find in this case...\[|x+y|=-(x+y)=|x|+|y|\]Now, imagine we have \[{x\ge0\land{y\lt0}}\]\[|x+y|=x+y<x<|x|+|y|\]Similarly,\[{x\lt0}\land{y\ge0}\]\[|x+y|=x+y<y<|x|+|y|\]

Answer 3

thank youuuuuuu!