Prove that $|x+y| \leq |x|+|y|$

Question

How to Prove the triangle inequality which says for all x (no matter how big or small) and for all y (no matter its size) in the set of irrational+rational numbers, this holds: $|x+y| \leq |x|+|y|$

Answer 1

Notice

$$ - |x| \leq x \leq |x| $$ $$ - |y| \leq y \leq |y| $$

Adding up, we obtain

$$ -( |x| + |y| ) \leq x + y \leq |x| + |y| $$

this implies

$$ |x + y| \leq |x| + |y| $$

Answer 2

Hint: Use $|x|\geqslant x$ and $|y|\geqslant y$ and $|x|\geqslant -x$ and $|y|\geqslant -y$ together to prove that: $|x|+|y|\geqslant x+y$ and $|x|+|y|\geqslant -(x+y)$, then your result follows.