Question about one of the first problems in Spivak's Calculus

Question

It's about Chapter I, Problem 21 from Spivak's Calculus:

Prove that if:

$|x - x_0| < \frac{\epsilon}{2}$ and $|y - y_0| < \frac{\epsilon}{2}$

then

$|(x + y) - (x_0 + y_0)| < \epsilon$

$|(x - y) - (x_0 - y_0)| < \epsilon$

I tried somehow making $2\times$ the first expressions equal $1\times$ the second expressions, but ended up in a big hairy mess, not sure what to do. I feel real dumb for not being able to do this, it's just chapter 1. Anyone who's done this have a tip? Maybe try to represent absolute values as $\sqrt{x^2}$ everywhere?

Answer 1

Hint: The triangle inequality states that

$$|a + b| \le |a| + |b|$$

for all choices of $a$ and $b$. This is an extremely important inequality to know.