Proof if $|x-x_0| < \frac{\epsilon}{2}$, $|y-y_0| \lt \frac{\epsilon}{2}$, then $|(x-y) - (x_0 - y_0)| \lt \epsilon$

Question

I can show the case where $|(x+y) - (x_0 + y_0)| \lt \epsilon$. For the minus case, I tried: $$\begin{align} |(x-y)-(x_0-y_0)| &= |(x-x_0) - (y-y_0)| \\ &\geq |x-x_0| - |y-y_0| \end{align}$$

But I don't know how to proceed from here.

Answer 1

By the triangle inequality $$\vert x-x_0 -(y-y_0) \vert \leqslant \vert x-x_0\vert + \vert y-y_0 \vert< \frac \epsilon 2 + \frac \epsilon 2=\epsilon . $$