# If $T\subset\mathbb{R}$ is bounded and $S \subset T$, then $\sup S \leq \sup T$ and $\inf T \leq\inf S$

Let $S$ and $T$ be nonempty sets of $\mathbb{R}$, with $T$ a bounded set and $S \subset T$. Prove that $\sup S \leq \sup T$ and $\inf T \leq\inf S$.

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## 1 Answer

We have $\inf T \le t \le \sup T$ for all $t \in T$. Since $S \subset T$, if $s \in S$, then we have $\inf T \le s \le \sup T$. It follows that $\inf T \le \inf S$, and $\sup S \le \sup T$.

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