Suppose I have a sequence of functions $f_n: X \to Y$ between metric spaces that converges uniformly to $f$. Then if I change the metric on $Y$ to a Lipschitz equivalent metric, the sequence of functions will still converge uniformly to $f$.
However, I am told that if I change the metric on $Y$ in such a way that does not disturb the induced topology, then $f_n$ may no longer converge uniformly. My question is: is there a simple example of a sequence of functions from $X$ to two homeomorphic metric spaces $Y$, $Y'$ converging uniformly in one but not the other?
EDIT: question modified slightly in response to comments.