The Deranged Mathematician: Repeat Ad Nauseum

I learned induction from Larry Gerstein's Introduction to Mathematical Structures and Proofs , which had an... unusual illustration for it. (As a side note: I think this is a measure of how it is getting harder to find things on the Internet. If I remember rightly, just a few years ago I searched for this illustration on Google and was able to find it. It is no longer possible if you don't remember the exact book! I even tried to use my Copilot to narrow down the search, and it insisted that what I described couldn't have come from a book and must have been something like a handout.) Side note aside, what can I say about this brief introduction to induction? My goal in this article was to both give the basic examples that one needs to start, where the inductive hypothesis is essentially given to you, along with some more advanced examples of how one might actually use it in practice. I.e., you might need to discover the inductive hypothesis on your own, and this requires some work! The simplest case is that you might gather numerical evidence until you figure out the pattern. But I also have an example (involving getting a bound on the n -th prime) which takes a more circuitous road: you must first generalize the problem, then search for numerical evidence to get a conjecture for what is true... and then preferably simplify it a little until you get down to something provable. You'll see what I mean---it ends up using just about every big-picture technique that I have covered. Read the full post (for free) on Substack: Repeat Ad Nauseam