- #1
Steve Turchin
- 11
- 0
Homework Statement
Prove that if
## |x-x_0|<\min (\frac {\epsilon}{2(|y_0|+1)},1)## and ##|y-y_0|<\frac{\epsilon}{2(|x_0|+1)} ##
then
## |xy-x_0y_0|<\epsilon ##
Homework Equations
N/A
The Attempt at a Solution
From the first inequality I can see that:
## |x-x_0|<\frac {\epsilon}{2(|y_0|+1)} ## and ## |x-x_0|<1 ##
From the first and second inequalities:
## |x-x_0|(2(|y_0|+1))<\epsilon ##
## |y-y_0|(2(|x_0|+1))<\epsilon ##
so by adding up both inequalities I get:
## |x-x_0|+|y_0|\cdot|x-x_0|+|y-y_0|+|x_0|\cdot|y-y_0|<\epsilon ##
and now I know that I need to write ##xy-x_0y_0## in a way that involves ##x-x_0## and ##y-y_0##
Thanks in advance!