There is a completely different approach to this problem which has gained some notoriety. Often one wants multiple approaches in order to understand a problem more deeply; here, we have picked a geometric approach.

It happens that Don Zagier has provided the culmination of a series of proofs using only sets and functions, and that proof takes only one line to write down! This is reproduced from a famous article with the following title.

In the exercises you will be asked to verify the various statements that this proof depends on. Although perhaps it is not a one-liner after all, it is still fun.