Proposition

Let $X$ be a discrete topological space. Show that the only convergent sequences in $X$ are the ones that are eventually constant, that is, sequences $(x_i)$ such that $x_i = x$ for all but finitely many $i$.

Solution

It is easy to see that eventually constant sequences converge. Let $(x_i)$ be a convergent sequence in $X$. Let $x$ be a limit. Then the singleton ${ x }$ is a neighborhood of $x$, so there must exist an $N \in \mathbb{N}$ such that $\forall n \geq N, x_n = x$. Therefore, $(x_i)$ is eventually constant.