I have a question about the proof listed here.
The proof is simple enough in that it begins with assuming that $\pi$ can be written as a fraction of two numbers, constructing a function on the basis of those two numbers and then finding two contradictory properties of this function.
Now comes the question: How could I have come up with this proof?
As a second year math student I have to give a small talk about this proof. It would seem interesting if after proving it I could explain a bit about how Niven could have come up with this proof. The second part (where the function is shown to lie between 1 and 0) seems easy. For the first part he seemingly defines functions out of the sky and then those all neatly work together to show that the integral must have an integer as a value.
What I have found is that it is logical that $\sin(x)$ is involed, as otherwise no known property of $\pi$ would be used in the proof.
