Prove that if $$|x-x_0| < \min\left(\frac{\epsilon}{2(|y_0| + 1)}, 1\right)$$ and $$|y-y_0| < \frac{\epsilon}{2(|x_0| + 1)},$$ then $|xy - x_0y_0| < \epsilon.$
I am doing some problems in Spivak's Calculus on inequalities and came across this problem. Currently I have a sketch of a solution that breaks down the problem into many cases and it is kind of long and messy. I thought maybe someone here could provide a clean and easier solution? If there is a nice solution please tell me a bit behind the thought process (like how you came up with it), instead of giving it as it is.