While idly browsing I came across a rather nice paper of Zhou and Markov (2009), which led to the first time I have ever learnt a proof of the famous and slightly random fact that is irrational.

The idea is to construct a sequence of expressions involving integrals, which are on the one hand (integrating by parts) given by a recurrence relation which guarantees that they are positive integers, but which on the other hand must clearly tend to zero, and of course this gives a contradiction. The rabbit from the analyst’s hat seems to be to consider the functions and then integrate through by a sine function. This then very quickly just works. Ok, let’s do it.

Suppose , and consider . However, we will also show that the are positive integers. Indeed, and for , integrating by parts gives that

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