I had posted a portion of this earlier asking about how to interpret min(). I received some excellent answers, however, I have run into problems and feel stuck. I am posting the question in its entirety.
Let $ x, x_{0},y,y_{0} $ be real numbers and $ \varepsilon $ be a positive real number. If we have
$|x-x_{0}|<\min \left(\frac{ \varepsilon }{2(|y_{0}|+1)}, 1\right) \text{ and } |y-y_{0}|< \frac{ \varepsilon }{2(|x_{0}|+1)},$ then prove that $ |xy-x_{0}y_{0}|<\varepsilon $.
A hint is provided:
Write $xy-x_{0}y_{0}$ in terms of $x-x_{0} $ and $y-y_{0} $ and use the triangle inequality twice.
I've been rearranging and writing out what I know etc. in an attempt to find a solution:
$ |x-x_{0}|(2|y_{0}|)+2|x-x_{0}|<\varepsilon $
$ |y-y_{0}|(2|x_{0}|)+2|y-y_{0}|<\varepsilon $
$ |x-x_{0}|< 1 $