I am trying to understand the proof of "Supremum of lower bounds is infimum":
Let A be bounded below, and define B = {b in R : b is a lower bound for A}. Show that sup B = inf A.
In particular, I have shown that $\alpha = \sup B$ exists so far, and want to show $\alpha = \inf A$. However, that means $(\forall a \in A, a \geq \alpha)$ However, I do not understand why that's necessarily true. I know that $\forall a \in A, b \in B, b \leq a$. However, $\alpha$ doesn't necessarily be in $B$, and thus might not satisfy that? Am I mistaken somewhere, or is this by definition of something?
Thank you.
HINT: Suppose that there is an $a\in A$ such that $a<\alpha$. Now remember that $\alpha=\sup B$, so there must be a $b\in B$ such that ...?