Hi LCKurtz

I guess I might have been avoiding red question. I don't know what you mean "put in the overestimates." Do you mean to actually
substitute ##|x-x_0|## with ##\frac{\epsilon}{2|y_0|+1}## ?
If so, I did not really know you could do that with inequalities. Is that what you meant though?
Here is what I have so far to summarize:
<br />
\begin{array}{c}<br />
|xy-x_0y_0| &=& |xy-x_0y_0 + xy_0 - xy_0| \qquad\qquad\qquad (1) \\<br />
&=& |x(y-y0) + y_0(x-x_0)| \qquad\qquad\qquad (2) \\ <br />
&<& |x(y-y0)| + |y_0(x-x_0)| \qquad\qquad\qquad (3) \\<br />
&<& \frac{|y_0|\epsilon(2|x_0|+1) + |x|\epsilon(2|y_0|+1)}{(2|y_0|+1)(2|x_0|+1)} \qquad(4)<br />
\end{array}<br />
Edit OK. I see that I kind "substitute" them in by transitivity. Just had to convince myself. I added it in as line (4). It would seem that if I could prove that the RHS of (4) is less than ##\epsilon## I could finish. So now I need to see if:
\frac{|y_0|\epsilon(2|x_0|+1) + |x|\epsilon(2|y_0|+1)}{(2|y_0|+1)(2|x_0|+1)} \stackrel{?}{<} \epsilon \qquad\qquad(5)
Is this going in the right direction?