Consider the topology on $\mathbb{R}$ in which a set is open if $U = V \setminus C$, where $V$ is open in the usual topology and $C$ is a countable set.
Prove that in this space a sequence converges iff it is eventually constant.
Consider the topology on $\mathbb{R}$ in which a set is open if $U = V \setminus C$, where $V$ is open in the usual topology and $C$ is a countable set.
Prove that in this space a sequence converges iff it is eventually constant.