I am trying to do the following exercise from Hirsch's Differential topology :
Let $p:M\rightarrow N$ a $C^s$ map and $f:N\rightarrow M$ a $C^r$ section of $p$. If $1\leq r<s\leq \infty$ then $f$ can be approximated by $C^s$ sections. In fact if $g$ is sufficiently $C^r $ close to $f$ we have that $g(N)$ is the image of a section.
There's a first thing I need to clarify, these are suppose to be sections of $p$ right ? That is the approximations $g_n\rightarrow f$ have to satisfy $p \circ g_n =Id_N$?
Now my first idea when trying to prove this is that well since we know that $C^s(N,M)$ is dense in $C^r_S(N,M)$ then let's just show that the set of $C^r$ maps that are sections of $p$, for $r\geq 1$, is open in the strong topology. This does not seem to be at all True, so I gave up on this idea.
Then I though well we can approximate $f$ by $C^s$ functions $g$ and this also means, it's an exercise in Hirsch, that there exists $m\in \mathbb{N}$ and a compact set such that if $n>m $ then $g_n(x)=f(x)$ for all $x\in N-K$. And so these $g_n$ would be sections for $p$ outside the compact set, but now I can't come up with anything that fixes the issue inside the compact set.
Also I haven't been able to figure out why we need that $r\geq 1$, and why tis can be dropped if we assume that $r=0$ and $p$ is submersive.
New edit : I guess another idea would be to take $C^s$ functions $g_n$ such that $g_n \rightarrow f$, and so since the composition is continuous we would have that $p\circ g_n \rightarrow Id$, and now since diffeomorphisms are open we would get a subsequence such that $p\circ g_{n_k}$ is a diffeomorphism and we could consider the functins $g_{n_k}\circ (p\circ g_{n_k})^{-1}$ that will be sections of $p$ but I am not very sure that they will be convergent to $f$, maybe this is where it enters the fact that since $r\geq 1$ we would have that $p$ is submersive in an open set that contains the image of $f$, I am not intirely sure.
Any help with this is aprecciated.
Thanks in advance.