Let $M,N$ be two smooth manifolds. $f: M\rightarrow N$ is an immersion. $L$ is closed in M and $f:L\rightarrow f(L)$ is a homeomorphism, implying $f(L)$ is closed in $N$. Prove that there exists a open neighborhood $V$ of $L$ such that $f|_V$ is injective.
I just learned the concept of immersion, submersion and embedding. I know that locally, an immersion is injective. Is this related to the problem? I also tried to use local charts to cover $L$ but have no idea how to continue.
Appreciate any help or hint!