Pointwise convergence implies uniform convergence in discrete metric space $(X,d)$?

Question

In general, does this hold for a sequence of functions in an arbitrary $X$? For a sequence to converge in the discrete metric, the sequence needs to become a constant sequence for a sufficiently large $n$. It seems to me that this would require every pointwise convergent sequence to also be uniformly convergent. I have not been able to come up with a counterexample. Any help would be appreciated.

Answer 1

This is only true if $X$ is finite. For instance consider $X=\mathbb{N}$ and the sequence of functions $$ f_n(x) = \begin{cases} 1, &\text{if $x=n$} \\ 0, &\text{else}. \end{cases}$$ Then $f_n$ converges pointwise to $f\equiv 0$, but for all $n\in\mathbb{N}$ $$d(f,f_n) = \sup_{x\in X} |f_n(x)-f(x)| = 1.$$