Let $S$ and $T$ be nonempty subsets of $\mathbb{R}$ with the following property: $s \leq t$ for all $s \in S$ and $t \in T$. Prove that $\sup S \leq \inf T$.
I have a question about this. We know that $\sup S \geq s$ for all $s \in S$, and $\inf T \leq t$ for all $t \in T$. But $\sup S$ can be greater than $\inf T$. How can I prove this?