I've been stuck on this problem for over a day, and the answerbook simply says "see chapter 5" for problems 20,21, and 22. But I want to complete the problem without using knowledge given later in the book, so I've been banging my head against the wall trying all sorts of things, but nothing I do seems to lead me anywhere.
The problem is as follows:
Prove that if
$|x - x_0| < min (\frac{\varepsilon}{2(|y_0| + 1)}, 1)$ and $|y - y_0| < min (\frac{\varepsilon}{2(|x_0| + 1)}, 1)$
then
$|xy - x_0 y_0| < \varepsilon$.
Here are some of the things I've been thinking about, I don't know which of these are useful (if any), but they somewhat outline the logic behind my various attempts.
Since at most $|x - x_0| < 1$ and $|y - y_0| < 1$ then it follows that $(|x-x_0|)(y-y_0|) < |x-x_0|$ and $(|x-x_0|)(y-y_0|) < |y-y_0|$
Also, $(|x - x_0|)(|y_0| + 1) < \frac{\varepsilon}{2}$ and $(|y - y_0|)(|x_0| + 1) < \frac{\varepsilon}{2}$ so $(|x - x_0|)(|y_0| + 1) + (|y - y_0|)(|x_0| + 1) < \varepsilon$. and since $|a + b| \leq |a| + |b| < \varepsilon$ I've tried multiplying things out, and then adding them together to see if anything cancels, but I can't make anything meaningful come out of it.
Also since $|a - b| \leq |a| + |b|$ I've also tried subtracting one side from the other, but to no avail.
I was also thinking that since $(|x-x_0|)(y-y_0|) < |x-x_0|$, then I could try something along the lines of $(|x-x_0|)(y-y_0|)(|x_0| + 1) + (|x-x_0|)(y-y_0|)(|y_0| + 1)< \varepsilon$ and various combinations as such, but I just can't seem to get anything meaningful to come out of any of these attempts.
I have a sneaking suspicion that the road to the solution is simpler than I'm making it out to be, but I just can't see it.
