Limit proof, absolute value problem.

Hi,

I am reading a proof (in Spivak) that asserts that.

Given and , and given then that .

We choose and the mid point between the two limits for epsilon. .

The proof starts thus:

He continues to prove a contradiction...

I have a problem at this point. We are substituting epsilon back in which seems simple. but does for all possible values of any function. Is the sign reversible even when it's a function. This seems to be required to get to the last step and I am unconvinced that it always holds. Am I missing something else?

Thanks

Regards

Craig.