I have a question from the answer to this question. It shows that if $f_n: M \rightarrow N$ is a sequence of (Riemannian) isometries that converge uniformly to a map $f: M \rightarrow N$, then $f$ is a Riemannian isometry. To do so, we use the fact that Riemannian isometry and distance isometry coincide.
In the proof, Prof. Lee's hint tells us that it is enough to show $f(M)$ is closed. However, from here how do we conclude that $f(M) = N$? Do we have any knowledge that $f(M)$ is open? Any insights into this will be appreciable.