Limit proof, absolute value problem.
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?