Here is the question:
Let $X$ and $Y$ be metric spaces.Suppose a sequence of uniformly continuous maps $f_n : X \rightarrow Y$ converges uniformly to a map $f: X \rightarrow Y$.Does that imply that f is continuous ? Uniformly continuous ($X$ is not necessarily compact)?
I proved (hopefully right) that $f:X \rightarrow Y$ is continuous , even without using uniform continuity of $f_n$ but just assuming $f_n$ are continuous. I can't prove or disprove that $f$ is uniformly continuous and whether adding the compactness condition would change the answer.