User:IssaRice/Formalizing adjustment of epsilons

I want to elaborate on a thing that shows up when proving part (a) of https://taoanalysis.wordpress.com/2020/05/24/exercise-5-6-1/

Actually, a similar thing appears in Tao's original proof (Proposition 5.5.12) and the situation is simpler there (fewer variables to deal with), so let's start there.

Tao's proof

In the first part of the proof, we suppose $x^2<2$, and let $0<\epsilon <1$. Then we show the bound $(x+\epsilon {)}^2\le x^2+5\epsilon$. Now we say that since $x^2<2$, we can choose $0<\epsilon <1$ small enough that $x^2+5\epsilon <2$.

What's so sneaky about this? It's the fact that we say "let $0<\epsilon <1$" and then later adjust the value of $\epsilon$. Typically in math, when we say "let $0<\epsilon <1$", we are allowing the value to be arbitrary, i.e. potentially adversarially chosen. We can't let the adversary choose something, then later change our mind and say "wait a minute, let me pick it instead"!