A property of first-countable spaces

Wikipedia says that one of the most important properties of first-countable spaces is that given any subset $A$ of a topological space $(X,\tau)$ we have the following equivalence : $x\in \bar A \iff \exists(a_{n})_{n\in \mathbb{N}}\in A^{\mathbb{N}} : lim\ a_{n}=x $. So without this property in a topological space the equivalence fails more precisely the part (P) $x\in \bar A \implies \exists(a_{n})_{n\in \mathbb{N}}\in A^{\mathbb{N}} : lim\ a_{n}=x$. Now if we consider the space $(\mathbb{R},\tau_{C})$ where $\tau_{C}$ is the cofinite topology then this space is not first countable, yet I can't seem to find an example where (P) fails since in the cofinite topology if $x_{n}$ is a sequence then any point in the space is a limit of $x_{n}$.

The Wikipedia page says that a first countable space is Fréchet-Urysohn, which is a fancy name for the property that the closure of set is precisely the set of possible sequential limits from that set.

As it happens, the cofinite topology on an uncountable set is an example of a Fréchet-Urysohn space that is not first countable (so this desirable property is not equivalent to being first countable). So there no example exists.

A better example in this vein is the co-countable topology on say $\Bbb R$. Also not first countable but as any convergent sequence is eventually constant certainly not Fréchet-Urysohn as well.