Convergent Sequence in Set of Integers/Corollary

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Theorem

Let $\left \langle {x_n}\right \rangle_{n \in \N}$ be a sequence of distinct terms in the set $\Z$.

Then $\left \langle {x_n}\right \rangle_{n \in \N}$ is not convergent.


Proof

By Convergent Sequence in Set of Integers, $\left\langle{x_n}\right\rangle_{n \in \N}$ is convergent iff:

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

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

The result follows.

$\blacksquare$