Here is my question. $S$ is an ordered field and $E \subseteq S$ is bounded below, and $\beta$ is a lower bound of E. Prove $\beta = \inf E\ $ iff for every $\epsilon>0$ there exists a $y \in E$ such that $y<\beta + \epsilon$. Here is my attempt:
Proof ($\implies$): Suppose $\beta = \inf E$. Then set $a = \beta + \epsilon$ for any $\epsilon>0$. Then since $\beta$ is an infimum, if $\beta<a$ then $a$ is not a lower bound, which means there exists a $y \in E$ such that $y<a$. Substituting $a$ we get $y < \beta + \epsilon$.
I haven't tried the other direction yet, but I was wondering if this is a correct start, and also will the fact that $S$ is an ordered field rather than $\mathbb{R}$ affect the proof?
Your proof is correct. The point of not asumming $S=\mathbb{R}$ is that $E$ won't necessarily have an infimum (despite being bounded below), so you are proving that if you have the epsilon condition then the infimum does exist (and equals $\beta$).