I think the converse of the sequence lemma cannot hold without the property of being first countable, but I couldn't find any counterexamples.
The sequence lemma:
Let $X$ be a topological space, $A⊆X$ any subset and $x∈X$. If there is a sequence of points in $A$ converging to $x$, then $x∈\text{cl}(A)$; the converse holds if $X$ is first-countable.