$\newcommand{\o}{\mathcal{O}}\newcommand{\d}{\,\mathrm{d}}$I believe what I was required to show is a general version of the Peano existence theorem.
Let $\o\subseteq\Bbb R\times\Bbb R^n$ be open and contain the point $(x_0,y_0)$. Let $g:\o\to\Bbb R^n$ be continuous. Show that there exists some open neighbourhood $I$ of $x_0$ and a function $f:I\to\Bbb R^n$ satisfying $f’(x)=g(x,f(x)),\,f(x_0)=y_0$.
I have come up with a proof that I’d really like advice on: is it correct? If not, could someone hint how to improve it? I was supposed to use Lipschitz mappings and the Arzela-Ascoli theorem.
Let $\|y\|$ denote $\max_{1\le k\le n}|y_k|$. As $\o$ is open in Euclidean space, there is are $a,b\gt0$ with $R=[x_0-a,x_0+a]\times\prod_{k=1}^n[y_{0_k}-b]\subset\o$. As $g$ is continuous and $R$ is compact, $g(R)$ is compact in Euclidean space and therefore there is an $M\gt0$ such that $\|g\|\lt M$ on $R$. Let for any $h\gt0$ $I_h$ denote the open interval in $\Bbb R$ centred on $x_0$, radius $h$, and let $X_h$ denote the space of $\Bbb R^n$-valued continuous functions on the closure of $I_h$ that satisfy $\|f-y_0\|\le b$. It can be shown that since $X_h$ is a closed subspace of the compact metric space $\mathcal{C}(\overline{I_h})$, $X_h$ is a compact metric space (under the uniform metric).
For an arbitrary $\nu\in X_h$, which is clearly nonempty, put $\nu_1:x\mapsto y_0+\int_{x_0}^x g(t,\nu(t))\d t$. This is well defined as such $\nu$ are continuous and their image is contained in the domain of $g$, and the continuous composition is integrable. $\|\nu_1(x)-\beta\|\le|x-x_0|M\le Mh$, so $\nu_1\in X_h$ if $h\le\frac{b}{M}$ and $\nu_1$ is Lipschitz continuous by the same bounds with $M$. Inductively put $\nu_n:x\mapsto y_0+\int_{x_0}^x g(t,\nu_{n-1}(t))\d t$ for natural $n\gt1$. All of these functions are contained in $X_h$ if the condition on $h$ above is satisfied and they are all Lipschitz continuous with bound $M$. It follows that they are all equicontinuous and bounded functions on a compact metric space, and thus the Arzela-Ascoli theorem guarantees the existence of a subsequence $f_n=\nu_{n_k}$ which converges uniformly to a continuous function $f$ in $X_h$.
Taking limits, and acknowledging that the integral satisfies the dominated convergence theorem and the continuity of $g$, finds: $$f(x)=y_0+\int_{x_0}^x\lim_{n\to\infty}g(t,f_n(t))\d t=y_0+\int_{x_0}^x g(t,f(t))\d t$$
That is, there is a function $f$ on $I_h$ an open neighbourhood of $x_0$ which solves the IVP.
Is this right?