The problem states that:
[math]y_0 \neq 0[/math]
[math]|y - y_0| < \frac{\epsilon|y_0|^2}{2}[/math]
And I must use those to prove that:
[math]y \neq 0[/math]
[math]|\frac{1}{y} - \frac{1}{y_0}| < \epsilon[/math]
My professor told me to utilize the inverse triangle inequality:
[math]|a| - |b| \leq |a - b|[/math]
Solving the first part was easy - I changed one of the expressions and used the triangle inequality: I change [math]|y - y_0|[/math] to [math]|y_0 - y|[/math]
Then I use the inequality: [math]|y_0 - y| \geq |y_0| - |y| < \frac{|y_0|}{2} => -|y| < \frac{|y_0|}{2} - |y_0| => -|y| < -\frac{|y_0|}{2} => |y| > \frac{|y_0|}{2}[/math] So we know [math]|y| > 0[/math]
Now for the second part, I put the expression [math]|\frac{1}{y} - \frac{1}{y_0}|[/math] into the form [math]|\frac{y_0 - y}{yy_0}|[/math] , but I don't know what else to do.Can someone help me?
