Let $S$ be a set that is bounded below. Prove that a lower bound $w$ of $S$ is the infimum of $S$ if and only if for any $ϵ > 0$ there exists $t$ belonging to $S$ such that $t < w + ϵ$.
I started by saying $\inf S=w \le t$. Let $t$ be $ϵ$ close to $w$ so $t+ϵ=w$ which implies that $t<w+ϵ$.
I am not sure if what I've done is correct and I am even more confused on how to prove the other way ($\le$)
Best Answer
The infimum is defined to be the largest lower bound of a set. If a lower bound $w$ of $S$ is not the infimum of $S$, then there is a larger lower bound. Can you then find a suitable $\epsilon$? And conversely, when an $\epsilon$ with the given properties is given, can you find a lower bound of $S$ that is larger than $w$?