I am struggling with this real analysis problem.
Let $F$ be a non-negative continuous real-valued function defined on the infinite strip {$(x,y): 0 \leq x \leq 1, y \in \boldsymbol{R^1}$} with the property that $F(x,y)\leq 4$ for all $(x,y)\in [0,1]\times[0,2]$. Let $f_n$ be a continuous piecewise-linear function from $[0,1]$ to $\boldsymbol{R^1}$ such that $f_{n}(0) = 0$, $f_n$ is linear on each interval of the form $[\frac{i}{n},\frac{i+1}{n}]$, $i=0,1,...,n-1$ and for $x\in (\frac{i}{n},\frac{i+1}{n}),$ $ f'_{n}(x)=F(\frac{i}{n},f_{n}(\frac{i}{n})).$ Prove that there is a subsequence {$f_{{n}_{k}}$} of {${f_n}$} such that $f_{{n}_{k}}$ converges uniformly to a function $f$ on $[0,1/2]$.
I think I am suppose to use the Weierstrass theorem, but I am not sure of how to get started or if this applies to the theorem.
Hint
Use the Arzelà–Ascoli theorem theorem.
The sequence $\{f_n\}$ is uniformly bounded by $4$ as $f_n(0)=0$ and $0 \le f_n^\prime(x) \le 4$ for all $x \in [0,1]$.
It is also equicontinuous as each $f_n$ is piecewise linear with a bounded slope.