A problem regarding the triangle inequality

Question

I wonder how can i solve this problem? I know it uses the triangle inequality and adding and subtracting the same variable. Yet i can't seem to get the proof right.

Question:

Let $ \epsilon \gt 0 $

If $ | x-a | \lt \epsilon $ and $ | y-b | \lt \epsilon $

Then:

$ | xy-ab| \lt \epsilon(|a|+|b|+\epsilon) $

Thank you.

Answer 1

$|xy-ab|=|x-a|\cdot|y-b|+|y-b|\cdot |a|+|x-a|\cdot |b|<\epsilon^2+\epsilon \cdot |a|+\epsilon \cdot |b|=\epsilon(|a|+|b|+\epsilon)$

Answer 2

$|xy-ab|=|xy-ya+ya-ab|=$

$|y(x-a)+a(y-b)|<$

$<|y||x-a|+|a||y-b|<(b+|y-b|)|x-a|+ |a||y-b|<$

$(b+\epsilon)\epsilon+a\epsilon=\epsilon(a+b+\epsilon)$