I am trying to show that closed immersions are open in the strong topology $C_S^r(M,N)$. I know that a proper map $f$ into a locally compact haussdorf space, which $N$ is , will be closed . So now I am just trying to see that if I have a closed immersion between manifolds then it will be a proper map. If I have this then I have that the set of closed Immersions is just the intersection of the open set of Immersions with the one of proper maps and so it is open.
So does anyone know why a closed immersion is a proper map ? I have tried seeing this using subsequences but I got nowhere, so any hints are aprecciated.
Thanks in advance.