Peano's Existence Theorem: Questions about the proof

Question

Since there are already quite a few questions about the Peano Existence Theorem out there, I would still like to post the version from my textbook here inorder to grasp it's steps.

Let $G\subset\mathbb{R}^{n+1}$ open$, (t_0,x_0)\in G, f:G\rightarrow\mathbb{R}^n$ continuous. We are given the IVP $\hspace{3cm}\dot{x}=f(t,x),\hspace{0,5cm} x(t_0)=x_0.\hspace{3cm}(6.1)$

---

Peano's Existence Theorem

Let $G\subset\mathbb{R}^{n+1}$ open $(t_0,x_0)\in G, f:G\rightarrow\mathbb{R}^n$ continuous. Then there exists $\delta>0$ and a function $x\in C^1(J_{\delta};\mathbb{R}^n)$, $J_{\delta}=[t_0-\delta,t_0+\delta]$, such that $(t,x(t))\in G$ for all $t\in J_{\delta}$ and $x=x(t)$ solves $(6.1)$ on the interval $J_{\delta}$.

Proof. Let $\delta_0$ and $r>0$ fixed, such that $J_{\delta_0}\times\bar{B}_r(x_0)\subset G$. Set $M:=\{|f(t,x)|:(t,x)\in J_{\delta_0}\times\bar{B}_r(x_0)\}$. First we approximate $f$ uniformly on compact subsets of $G$ by a sequence of $C^1$-functions $f_k\in C^1(G,\mathbb{R}^n)$, these fucntions are then locally lipschitz in $x$. W.l.o.g let $|f_k(t,x)|\leq M+1$ for all $(t,x)\in J_{\delta_0}\times\bar{B}_r(x_0)$ and $k\in \mathbb{N}$. By the Picard-Lindelöf Theorem the IVP's

$\hspace{3cm}\dot{x}=f_k(t,x),\hspace{0,5cm}t\in J_{\delta},x(t_0)=x_0,$

have unique solutions $x_k\in C^1(J_{\delta};\mathbb{R}^n)$ on a common interval of existence $J_{\delta}=[t_0-\delta,t_0+\delta]$ and the values of the solutions stay in $\bar{B}_r(x_0)$, where $r>0$ is chosen like in the first sentence of the proof and $\delta=\text{min}\{\delta_0,r/(M+1)\}$. The sequence $(x_k)_{k\in\mathbb{N}}\subset C^1(J_\delta;\mathbb{R^n})$ is therefore uniformly bounded, but also equicontinuous because the derivatives $\dot{x_k}=f_k(t,x_k(t))$ are bounded by $M+1$. The Arzela-Ascoli Theorem therefore yields a on $J_\delta$ uniformly continuous subsequence $x_{k_m}\rightarrow x$. The $x_k$ abide the integral equations

$\hspace{3cm} x_k(t)=x_0+\int_{x_0}^{t}f_k(s,x_k(s)ds,\hspace{0,5cm} t\in J_\delta.$

After going over to the limit we get our desired solution.$\hspace{3cm}\square$

My questions now are the following:

1. First we approximate $f$ uniformly on compact subsets of $G$ by a sequence of $C^1$-functions. Why is this possible and how does it work ?
2. Why are these $C^1$-functions locally lipschitz in $x$?
3. Why is the fact that the derivatives are bounded an argument for equicontinuity?

Thanks in advance for taking your time.

Answer 1

1. Use the Stone-Weierstraß theorem to even get polynomial approximations. Or use convolution with a smooth kernel function (compact support, non-negative, integral $1$, the smaller the support the better the approximation).

2. $C^1$ implies locally Lipschitz, because any local bound on the derivative is a Lipschitz constant.

3. They are even equally Lipschitz continuous, as the bound on the derivative is a Lipschitz constant.