Spivak's proof that $\pi$ is irrational

Question

I'm reading chapter 16 of Spivak's Calculus, 4th edition, specifically proof that $\pi$ is irrational. Last part is unclear to me. He states that because \begin{equation} 0 < \pi a^n f_n(x) \sin \pi x < \frac{\pi a^n}{n!} \end{equation} then \begin{equation} 0 < \pi \int_0^1 a^n f_n(x) \sin \pi x\,dx < \frac{\pi a^n}{n!} \end{equation} the part that is unclear to me is how did we conclude that the integral also obeys inequality.

Answer 1

Integration satisfies that if $0<f<g$ for two functions, then $$0<\int_If<\int_Ig$$ for any interval $I$. Applying this in the above case gives the second inequality.

Answer 2

$(b-a)\min f\le\int_a^b f\le(b-a)\max f$.