Proving an Absolute Value Inequality

Question

I've just started reading Spivak's Calculus text (4th ed.) and am having some trouble on one of the exercises. The problem asks me to prove that if $|x-x_0|<\frac{\epsilon}{2}$ and $|y-y_0|<\frac{\epsilon}{2}$, then $|x-y-(x_0-y_0)|< \epsilon$. I've proven that it implies that $|x+y-(x_0 + y_0)|<\epsilon$ by adding the two given inequalities and using the addition triangle inequality for absolute value, but I can't find a way to apply the subtraction triangle inequality on this problem. Any help would be appreciated, thanks.

Answer 1

Hints:

$$|x-y-(x_0-y_0)|=|x-x_0-(y-y_0)|\le|x-x_0|+|y-y_0|\ldots$$