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In class the intermediate value theorem was given different to the theorem one usually encounters: https://en.wikipedia.org/wiki/Intermediate_value_theorem.

The Theorem is

$A\subseteq\mathbb{R}$ is connected iff for all $a,b\in\mathbb{R}$ if $a<c<b$, then $c\in A$.

I am not sure about the proof. I think the $\Rightarrow$ implication is proved by contrapositive:

Suppose there is $a,b\in\mathbb{A}$, such that there is some $c$ with $a<c<b$ but $c\notin A$. If we take $(-\infty,c]\cap A$ and $[c,\infty)\cap A$, both intersections are not empty, and the union equals $A$. So $A$ is not connected.

For $\Leftarrow$ I'm not sure how to proceed. I'll appreciate if someone has a solution.

Mangostino
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  • what is your definition of connected? – David P May 30 '22 at 22:17
  • If I understand correctly, what you're trying to prove is that connected subsets of the real line are intervals, and viceversa. If that's it you can take a look here or here – david_sap May 30 '22 at 22:20
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    Thank you for the reference. That is enough for me. I'll deleted if admins ask me to. But it think it will help for future seekers to find the posts suggested by @david_sap – Mangostino May 30 '22 at 22:21

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