The specific expression envolving epsilon on the third line of the proof seems like it came from hindsight. however, my question is, why does the end of the proof have to show that the difference is less than exactly epsilon? Wouldn't it have the same implications if i showed that it was less than 3$\epsilon$+1 or any random term evolving epsilon? Is it just for formality?
On
if you can get that the term is less than $p(\epsilon)$ with $p$ a polynomial such that $p(0)=0$ that is enough or even $$ p(\epsilon^a) $$ for some $a>0$ is enough.
I find proofs where they pick the multiples of $\epsilon$ at the beginning to make it work a bit artificial.
(see my book Proof Patterns for more discussion)
Quick answer: $3\varepsilon+1$ wouldn't be good, since it is not "small". But $3 \varepsilon$ would be fine, and moregenerally any fixed positive multiple of $\varepsilon$.
The choice of $\varepsilon / (2|b|+1)$ was made for convenience.