An Inequality Involving $min(x, y)$

An Inequality Involving $min(x, y)$

The following problem is from Spivak's Calculus (4th ed., pg. 18):



Prove that if:
$|x-x_0| < min(\frac{\epsilon}{2(|y_0|+1)}, 1)$ and $|y - y_0| <
\frac{\epsilon}{2(|x_0|+1)}$ then $|xy-x_0 y_0| <\epsilon$.



What I've done so far is to say that the since the first inequality
implies $|x-x_0| < 1$, combining this with the second inequality yields
$|(x-x_0)(y-y_0)|<\frac{\epsilon}{2(|x_0|+1)}$, or
$|(x-x_0)(y-y_0)|*(|x_0|+1) < \frac{\epsilon}{2}$. We also know from the
first inequality in the problem that
$|x-x_0|<\frac{\epsilon}{2(|y_0|+1)}$, so $|x-x_0|(|y_0|+1)<
\frac{\epsilon}{2}$. Combining yields: $|(x-x_0)(y-y_0)| *(|x_0|+1)
+|x-x_0|(|y_0|+1) < \epsilon$. I've tried manipulating the left hand side
to get $|xy-x_0y_0|$ with little success. Does this whole approach to the
problem seem ill-conceived? And if so, how can I rectify it? Or should I
try a totally different approach? Thanks.