Convergent Sequence in Set of Integers/Corollary

Theorem

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence of distinct terms in the set $\Z$.

Then $\sequence {x_n}_{n \mathop \in \N}$ is not convergent.

Proof

By Convergent Sequence in Set of Integers, $\sequence {x_n}_{n \mathop \in \N}$ is convergent if and only if:

$\exists k \in \N: \forall m \in \N: m > k: x_m = x_k$

But since $\sequence {x_n}_{n \mathop \in \N}$ is a sequence of distinct terms, for all $k \ne m$, one has $x_m \ne x_k$.

The result follows.

$\blacksquare$