I want to show the following: Let $\pi:M\to N$ be a submersion. Then, every point of $M$ is in the image of a smooth local section of $\pi$.
Since $\pi$ is a submersion, it is also an immersion because for linear maps injectivity, surjectivity and bijectivity are equivalent. So $M$ and $N$ have the same dimension. Also, $\pi$ has constant rank $k$. Now take a $p\in M$. By the constant rank theorem there exist smooth coordinates $(x^1,...,x^k)$ centered at $p$ and $(v^1,...,v^k)$ centered at $\pi(p)$ in which $\pi$ has the coordinate representation $\pi(x^1,...,x^k)=(x^1,...,x^k)$, that is, $\pi$ is locally the identity map. So locally we can take its inverse as a local section which satisfies the proposition.
I'm not really convinced about the identity map part, is everything correct?