Please help me with the following question:

Let $S \subset T \subset \Bbb{R}$, then

  • if $T$ is bounded from above, then $\sup S \le \sup T$

  • if $T$ is bounded from below, then $\inf T \le \inf S$

  • if $T$ is bounded, then $\inf T \le \inf S \le \sup S \le \sup T$.

Thanks for the support!


We are of course assuming that $S$ and hence $T$ is nonempty. Since $\sup T\ge x$ for all $x\in T$, we must have $\sup T\ge x$ for all $x\in S$. So, $\sup T$ an upper bound of $S$, hence $\sup S\le\sup T$, since $\sup S$ is the smallest upper bound of $S$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.