Convergence of Sequence in Discrete Space/Corollary

Corollary to Convergence of Sequence in Discrete Space

Let $T = \left({S, \tau}\right)$ be a discrete topological space.

Let $H = \left \langle{x_n}\right \rangle_{n \in \N}$ be a sequence in $S$.

Let $\left \langle {x'_n}\right \rangle_{n \in \N}$ be a sequence of distinct terms in $S$.

Then $\left \langle {x'_n}\right \rangle_{n \in \N}$ is not convergent in $T$.

Proof

By the definition of a sequence of distinct terms:

$\forall x \in \left \langle {x'_n}\right \rangle_{n \in \N}: r \ne s \implies x_r \ne x_s$

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.

$\blacksquare$