State and prove the Arzela-Ascoli theorem.

Let $N$ be a positive integer. Consider the subset $\mathcal{S}_{N} \subset C([0,1])$ consisting of all thrice differentiable solutions to the differential equation

$f^{\prime \prime}=f+\left(f^{\prime}\right)^{2} \quad$ with $\quad|f(0)| \leqslant N, \quad|f(1)| \leqslant N, \quad\left|f^{\prime}(0)\right| \leqslant N, \quad\left|f^{\prime}(1)\right| \leqslant N .$

Show that this set is totally bounded as a subset of $C([0,1])$.

[It may be helpful to consider interior maxima.]