If $\displaystyle S \subseteq \mathbb{R}$ is bounded below but not bounded above, then $\displaystyle S$ is either $\displaystyle \left( a,\infty \right] $ or $\displaystyle \left[ a,\infty \right) $ for some $\displaystyle a \in \mathbb{R}$.

I know from a previously proved theorem that if $\displaystyle S$ is a connected subset of $\displaystyle \mathbb{R}$ and $\displaystyle a<b$, $\displaystyle a,b \in S$, then $\displaystyle [a,b] \subseteq S$. And it seems intuitive that $\displaystyle a$ is the greatest lower bound for this set, but I am not sure how to begin to prove it.