I'm trying to prove the following:
"Let $\Omega\subset\mathbb{R}\times\mathbb{R^n}$ be an open set and $f\in C^k(\Omega), (x_0,y_0)\in\Omega$.
Show that the solutions of $y'=f(x,y)\ (x\in\mathbb{R},y\in\mathbb{R^n})$ belong to $C^{k+1}$."
My attempt:
By induction on $k.$
If $k=0$ then by Peano-Picard theorem there exists a solution $\phi\in C^1$ to the DE.
Suppose now that $f\in C^{k+1}(\Omega)$; then $f\in C^k(\Omega)$ so by inductive hypothesis there exists a solution $\phi\in C^{k+1}$ such that $\phi'=f(x,\phi(x))$ i.e. the derivative of $\phi$ is $C^{k+1}$ thus $\phi$ is $C^{k+2}$ and this closes the induction.
Can someone confirm that my proof is correct?
Thanks.