Greatest Lower Bound in $\mathbb{R}$ as a corollary of the LUB? I can assume as fact that $\mathbb{R}$ is an ordered field in which every non-empty subset that is bounded above has a least upper bound.

My question is whether I can also assume as fact that every non-empty subset that is bounded below also has a greatest lower bound. I'm trying to show that if $A,B \subseteq \mathbb{R}$, and $A,B \neq \emptyset$, then there exists a least upper bound for $A$ (I've proven this), and a greatest lower bound for $B$ (this is what I'm currently concerned with).

Any tips, or helpful definitions?
HINT: If $A\subseteq\Bbb R$ is bounded below, then $-A=\{-a:a\in A\}$ is bounded above.

Alternative HINT: Let $B$ be the set of lower bounds for $A$. $B$ is bounded above, and its supremum is ... ?
Yes, if every nonempty set that is bounded above has a least upper bound, then every nonempty set that is bounded below has a greatest lower bound. I know of two different ways to prove this.


Suppose $B$ is nonempty and bounded below. Then the set $-B=\{-x:x\in B\}$ is nonempty and bounded above, so it has a least upper bound, call it $u$. Then $-u$ is the greatest lower bound of $B$.
Never mind, just read Brian's answer!