There was a particular technique used in a topology course that was quite interesting. It was used to prove that that every closed interval \([a, b] \subseteq \mathbb{R}\) is connected and compact. It is centered on the result that every open subset of \(\mathbb{R}\) has a supremum. It goes like this:

Let \(P\) be the property we require \([a, b]\) to satisfy and define the following subset

\[\begin{align*} A := \{ y \in [a,b] \ : \ P \text{ holds for } [a, y] \} \end{align*}\]

We then show \(A \neq \emptyset\) and \(A \subseteq \mathbb{R}\) is open. Once we have done this, we can take its supremum \(t := \sup A\). Now, if \(t \in A\) and \(t = b\) then it follows that \(P\) holds for \(A\). The essential idea is that once we have \(P\) holds for \([a, y]\) then we need to show it can be extended to the entire subset.

For example, to prove \([a, b]\) is connected, we have …