Writing Better Proofs Part 2: Recap
Proof can rapidly get complicated or involve multiple cases and edge conditions. Once you have finished tying up all the loose ends, take the time to give a very quick recap of what you have actually done.
Using the Going Up theorem we can extend … and so \(\dim R \ge \dim S\).
On the other hand, if we have … and thus \(\dim S \ge \dim R\).
We have thus shown that \(\dim R \ge \dim S\) and \(\dim R\le \dim S\) and so these must in fact be equal.
This is particularly important if you are writing a proof using contradiction. Here, you should note what assumption the contradiction shows false and how that implies the result.
… but then \( 2\mid b\) and so \(a\) and \(b\) are not coprime.
This contradicts our initial assumption and thus \(\sqrt{2}\) cannot be written as a fraction of integers and thus is not rational.
This philosophy fits into the adage for making a good presentation:
Tell them what you are going to tell them.
Then tell them.
Then tell them what you told them.^{1}
Even if a proof isn’t a presentation, you want your reader to come away with a full understanding of the argument.
Best to leave them with the punchline.
This is part of a series on writing better proofs. The pages in the series are not static and will be updated and improved as time goes on. If you have comments or suggestions, please email me. I’d be particularly interested in hearing if you disagree with me or have suggestions for topics to cover.

This triplet has been at times attributed to Arsitotle, Dale Carnegie and J.H. Jowett of Birmingham. ↩︎