TITLE: Limit in a topology.
QUESTION [8 upvotes]: Let $\mathbb{Z}$ denote the set of integers. For $c$ and $r$ in $\mathbb{Z}$, define: $$B(c, r):=\{c+kr\ |\ k\in \mathbb{Z}\}.$$
As $c$ varies over all integers and $r$ over all positive integers, the sets $B(c, r)$ form a basis for a topology on $\mathbb{Z}$.
Does the following limit exist with respect to this topology? $$\lim_{n\to\infty}(n!-2)^2.$$
I have no idea how to think about this problem. Any hint or help.

REPLY [4 votes]: The limit is $4$.  For each $n \in \Bbb N$, the sequence element is $4$ plus a multiple of $n!$.  Any basis element that contains $4$ can be expressed in the form $\{ 4+ kr \mid k \in \Bbb Z \}$, which will contain all elements of the sequence with $n \geq r$.