Convergence of Sequence in Discrete Space/Corollary

Corollary to Convergence of Sequence in Discrete Space

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

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

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

By the definition of a sequence of distinct terms:

$\forall x \in \sequence {x'_n}_{n \mathop \in \N}: r \ne s \implies x_r \ne x_s$

Hence trivially:

$\neg \exists k \in \N: \forall m \in \N: m > k: x'_m = x'_k$

The result follows from Convergence of Sequence in Discrete Space.

$\blacksquare$