Let $S$ and $T$ be non-empty bounded sets of real numbers and suppose that $s<t$ for every $s \in S$ and every $t \in.T$.

Question

I'm doing a university Real-analysis course and I was hoping to get some feedback on a practice question I've been working on. Here's the question.

I believe i , ii and iii are all false. But I believe iv to be true because even though $s < t$ $Sup S$ or $inf T$ can lie on the border of some set, i.e. a bound that is not a maximum or minimum. Therefore $Sup S \in T$ or $infT \in S$ thus fulfilling $SupS \leq infT$. As for b. It changes to false for the reasons mentioned above.

Any feed back would be much appreciated!

Answer 1

Each $s\in S$ is a lower bound for $T$, each $t\in T$ is an upper bound for $S$. Hence $$\inf S\le \sup S\le \inf T\le \sup T $$ and all of(i), (ii), (iii), (iv) are true.

However, as the example $S=[-1,0)$, $T=(0,1]$ shows, $\sup S<\inf T$ need not be true.