there's example from the logic appendix from Tao's Analysis 1 Book with the aim of clarify how you can do proof of existence of an $\epsilon$ st a property $P(x)$ is true.
The thereom and the proof are:
I don't quite understand what the author meant with the phrase in green. It should have something to do with the fact we have to proof for the chosen $\epsilon$ that $P(x)$ is true for a infinite values of $x$ (inside the interval). But I am not fully understanding this.
And other thing: Note the author states that "if we had chosen $\epsilon$ to depend on $x$ and $y$ then the argument..." What does he meant by that? If my $\epsilon$ it's only function of $x$ then we would have no problem here?