Theorem Suppose $S$ is an ordered set with the least-upper-bound property, $B \subset S$, $B$ is not empty, and $B$ is bounded below. Let $L$ be the set of all lower bounds of $B$. Then $\alpha = \sup L$ exists in $S$, and $\alpha = \inf B$.
I have proved the existence of $\sup L = \alpha$.
Question I am not able to prove that $\alpha = \inf B$?
Rudin has provided the following explanation - "If $\gamma < \alpha$ then $\gamma$ is not an upper bound of $L$, hence $\gamma \notin B$. It follows that $\alpha \leq x$ for every $x \in B$. Thus $\alpha \in L$."
Question I am not able to follow this argument? Specifically, shouldn't $\alpha < x$ for every $x \in B$?
If $\gamma \lt \alpha$, then $\gamma$ is less than the least upper bound of $L$, hence $\gamma$ is not an upper bound of $L$. But since $L$ is the set of all lower bounds of $B$, we see that every $ x \in B$ is an upper bound of $L$, because every $x \in B$ is greater than any of the lower bounds of B. Since $\gamma$ is not an upper bound of $L$, $\gamma \not\in B$.
We also have stated that every $x \in B$ is an upper bound of $L$. Since $\alpha$ is the least upper bound of $L$, $\alpha \le x$ for all $ x \in B$.