Convergent Sequence in Set of Integers

Theorem

Let $\sequence{x_n}_{n \in \N}$ be a sequence in the set $\Z$ of integers considered as a subspace of the real number line $\R$ under the Euclidean metric.

Then $\sequence{x_n}_{n \in \N}$ converges in $\R$ to a limit if and only if:

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

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

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.

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

Consider the open set in $\R$:

$U := \openint{l - \dfrac 1 2}{l + \dfrac 1 2}$

Then $\forall x \in \Z: x \in U \implies x = l$

It follows by definition of convergence that:

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

where $x_k = l$.

Now suppose that:

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

Then trivially $\sequence{x_n}_{n \in \N}$ converges to the limit $x_k$.

$\blacksquare$