Topology and axiom of choice

Solution 1:

It is consistent with the ZF axioms that there is a dense set of reals $D\subset\mathbb{R}$ having no countable subset. Such a set is infinite, but Dedekind finite. It follows that any point in $\mathbb{R}-D$ is in the closure of $D$, but not a limit of any sequence from $D$, since any such sequence would give rise to a countable subset of $D$.

Meanwhile, your argument does not require full AC, but only countable AC, since you are making countably many choices of points closer and closer to $x$.

Solution 2:

Some papers: Disasters in metric topology without choice

Continuing horrors of topology without choice

and references therein.