Prove that if $\ |x-x_0| < \frac ε2$ and $|y-y_0| < \frac ε2 $ then $|(x+y) - (x_0 + y_0)| < ε$ and $|(x-y) - (x_0 - y_0)| < ε$

Question

This is an exercise of Spivak Ch. 1 Q20

Prove that if

$\ |x-x_0| < \frac ε2$ and $|y-y_0| < \frac ε2 $

then

$|(x+y) - (x_0 + y_0)| < ε$

$|(x-y) - (x_0 - y_0)| < ε$

So far I've just gotten

$|x-x_0| - |y-y_0| < ε$

How do I deal with things from here? Preferably not to just spell out the answer, I just need a direction.

Answer 1

Notice that

$|x-x_0+y-y_0|\le|x-x_0|+|y-y_0|<\epsilon$, by the triangle inequality.

Now you just need to associate some terms to have:

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

For the second inequality recall that $|y-y_0|=|y_0-y|$, thus

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