(a) For $K$ a compact Hausdorff space, what does it mean to say that a set $S \subset C(K)$ is equicontinuous. State and prove the Arzelà-Ascoli theorem.

(b) Suppose $K$ is a compact Hausdorff space for which $C(K)$ is a countable union of equicontinuous sets. Prove that $K$ is finite.

(c) Let $F: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be a bounded, continuous function and let $x_{0} \in \mathbb{R}^{n}$. Consider the problem of finding a differentiable function $x:[0,1] \rightarrow \mathbb{R}^{n}$ with

$$x(0)=x_{0} \quad \text { and } \quad x^{\prime}(t)=F(x(t))$$

for all $t \in[0,1]$. For each $k=1,2,3, \ldots$, let $x_{k}:[0,1] \rightarrow \mathbb{R}^{n}$ be defined by setting $x_{k}(0)=x_{0}$ and

$$x_{k}(t)=x_{0}+\int_{0}^{t} F\left(y_{k}(s)\right) d s$$

for $t \in[0,1]$, where

$$y_{k}(t)=x_{k}\left(\frac{j}{k}\right)$$

for $t \in\left(\frac{j}{k}, \frac{j+1}{k}\right]$ and $j \in\{0,1, \ldots, k-1\}$.

(i) Verify that $x_{k}$ is well-defined and continuous on $[0,1]$ for each $k$.

(ii) Prove that there exists a differentiable function $x:[0,1] \rightarrow \mathbb{R}^{n}$ solving (*) for $t \in[0,1]$.