Fake proof regarding continuity.

Question

So to my understanding the fallacy in this argument is that it's circular. Since he didn't declare an ϵ at the beginning of the proof, his choice of ϵ is dependant on δ, which in turn is dependant on ϵ. We were taught to always begin our proofs with "Let ϵ>0 ... ", or "We choose ϵ=[blank/fill in later]>0 ... " and the fact that he doesn't raised my suspicion. But I can't TRULY point out his mistake, and my thought regarding the fallacy is purely intuitive, and the fact that everything else looks fine really makes me wonder if there is something that I am missing..

Answer 1

The catch is that when it states that "there exists an $ \epsilon_0 > 0 $", this $ \epsilon_0 $ depends on $ x_1 $, it is $ \epsilon_0(x_1) $. But $ x_1 $ was chosen based on $ \delta_1 $ which in turn was defined based on the choice of $ \epsilon_0 $. So the first $ \epsilon_0 $ and the second are not the same positive number!

The second $ \epsilon_0 $ is equal to $ \epsilon_0 = \epsilon_0 (x_1(\delta_1(\epsilon_0))) $