Convergence of Sequence in Discrete Space

Theorem
Let $T = \struct {S, \tau}$ be a discrete topological space.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $S$.

Then $\sequence {x_n}_{n \mathop \in \N}$ converges in $T$ to a limit :
 * $\exists N \in \N: \forall n \in \N: n > N \implies x_n = x_N$

That is, the sequence reaches some value of $S$ and "stays there".

Sufficient Condition
Suppose $\sequence {x_n}_{n \mathop \in \N}$ converges to a limit $L$.

Then:

Necessary Condition
Let $N \in \N$ such that:


 * $\forall n \in \N: n > N \implies x_n = x_N$

Let $U$ be an open neighborhood of $x_N$.

Then:


 * $\forall n \in \N: n > N \implies x_n = x_N \in U$

Thus $\sequence {x_n}_{n \mathop \in \N}$ converges to $x_N$.