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 if and only if:

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

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

Corollary

Let $\sequence {x'_n}_{n \mathop \in \N}$ be a sequence of distinct terms in $S$.

Then $\sequence {x'_n}_{n \mathop \in \N}$ is not convergent in $T$.

Proof

Sufficient Condition

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

Then:

\(\ds \set L\)

\(\in\)

\(\ds \tau\)

Definition of Discrete Space

\(\ds \leadsto \ \ \)

\(\ds \exists N \in \N: \forall n \in \N: \, \)

\(\, \ds n > N \implies \, \)

\(\ds x_n\)

\(\in\)

\(\ds \set L\)

Definition of Convergent Sequence (Topology)

\(\ds \leadsto \ \ \)

\(\ds \exists N \in \N: \forall n \in \N: \, \)

\(\, \ds n > N \implies \, \)

\(\ds x_n\)

\(=\)

\(\ds L\)

Definition of Singleton

\(\ds \leadsto \ \ \)

\(\ds \exists N \in \N: \, \)

\(\ds x_N\)

\(=\)

\(\ds L\)

Universal Instantiation

\(\ds \leadsto \ \ \)

\(\ds \exists N \in \N: \forall n \in \N: \, \)

\(\, \ds n > N \implies \, \)

\(\ds x_n\)

\(=\)

\(\ds x_N\)

$\Box$

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 $x_N$.

$\blacksquare$