Understanding proof that $\pi$ is irrational Reading this: Simple proof that $\pi$ is irrational, I fail to understand the following part:

Since $n!f(x)$ has integral coefficients and terms in $x$ of degree
  not less than $n$, $f(x)$ and its derivatives (...) have integral
  values for $x=0$; also for $x=\pi=\frac{a}{b}$, since
  $f(x)=f(\frac{a}{b}-x)$

Assuming this, the rest I understand. But why is this true?

$n!f(x)$ is a polynomial with integer coefficients.
The derivative of a polynomial with integer coefficients is a polynomial with integer coefficents.
A polynomial with integer coefficients takes on integer values when evaluated at an integer.
