TITLE: How would I prove $|x + y| \le |x| + |y|$?
QUESTION [6 upvotes]: How would I write a detailed structured proof for:

for all real numbers $x$ and $y$, $|x + y| \le |x| + |y|$

I'm planning on breaking it up into four cases, where both $x,y < 0$, $x \ge 0$ and $y<0$, $x<0$ and $y \ge0$, and $x,y \ge 0$. But I'm not sure how I'd go about writing it formally.
Thanks!

REPLY [2 votes]: You might consider the following:
Consider squaring both sides since both sides are non negative (you don't have to worry about sign flips). Canceling yields
$$xy \leq |x||y|$$
which is true.