Prove the irrationality of $e=\exp(1)$ with $\exp $ defined as solution of ODE

Question

I am writing math lessons as I will teach soon on my freetime. In my course, I defined $\exp(x)$ as the solution to the O.D.E : $$f'(x)=f(x),\qquad f(0)=1$$ Is there a way to prove, maybe by contradiction, that $e$ is irrational without using series?

Answer 1

I don't think there is; I think you'll have to prove $e=\sum_{k\ge 0}\frac{1}{k!}$ first. However, there's an easy way to do that using the ODE. We make repeated use of $f(1)=1+\int_0^1 f(t) dt$: $$f(1)=1+\int_0^1 (1+\int_0^t f(t') dt')dt+\cdots$$So $$f(1)=1+\int_0^1 dt+\int_0^1 dt\int_0^t dt' t'+\cdots=\sum_{k\ge 0}\frac{1}{k!}.$$