Convergent sequences of integers are eventually constant.

So, I've been stuck on this question for the past hour, we haven't done one before in a lecture so I don't really no where to start, any help would GREATLY be appreciated!

So the full question is:

Assume $(a_n)_{n=1}^\infty$ is a convergent sequence of integers. Prove the existence of $N ∈ \mathbb{N}$ such that $a_i = a_j$ for all $i,j > N$

Honestly, the only thing I've done so far is writing down some notation and attempted to write it in a limit form, but I wasn't sure what it equaled, so I don't think that's the way to do it.

Thanks heaps again! :)

• Eventually they all must be close to the limit, right? What if you assume the limit is $x \in \mathbb{R}$? So eventually they all must be within $1/3$ of $x$. – Michael Mar 15 '17 at 5:06

$(a_n)$ is a Cauchy sequence, hence, for $\epsilon =1$ there is $N \in \mathbb N$ such that

$|a_i-a_j|<1$ for $i,j >N$.

Since $a_i-a_j \in \mathbb Z$, we have $a_i-a_j =0$ if $i,j >N$.

• This does not leave much to the imagination. – Michael Mar 15 '17 at 5:22

Since $\{a_n\}$ is a convergent sequence, $\{a_n\}$ is a Cauchy sequence. By definition, for any $\epsilon>0$, there exists $N_\epsilon \in \mathbb{N}$ such that

$$|a_i - a_j|<\epsilon \text{ for all } i,j \ge N_\epsilon$$

In particular, let $\epsilon = 0.1$, then there exists $N_{0.1}$ such that

$$|a_i - a_j| < 0.1 \text{ for all } i,j \ge N_{0.1}$$

However, since $\{a_n\} \subset \mathbb{N}$, all $a_i$'s have to be natural numbers.

In general, if $a_n \to a$, then for any $\epsilon>0$, the set of $n$ such that $a_n \notin B(a,\epsilon)$ is finite.

Hence the set of $n$ such that $a_n \notin B(a,{1 \over 2})$ is finite.

The set of integers is closed, hence $a$ is an integer, and so $B(a,{1 \over 2}) = \{a\}$.

In particular, there is some $N$ such that $a_n = a$ for $n \ge N$.

• Why the downvote? – copper.hat Mar 16 '17 at 13:19
• I mean, why the downvote Integrals? – copper.hat Mar 16 '17 at 13:40

This is the same answer as copper.hat except phrased in a notation that might be easier to understand for beginners (who would be asking this question in the first place).

Since $$(a_n)_{n\geq 1}$$ converges, there exists a limiting value $$a\in\mathbb Z$$. By definition of the limit, this means that for all $$\epsilon>0$$ there exists an $$N$$ such that $$|a_n-a|<\epsilon$$ for all $$n>N$$. In particular we may choose $$\epsilon=\tfrac12$$. Then for all $$n>N$$ we have $$|a_n-a|<\tfrac12$$. But since $$a_n,a$$ are integers, this implies that $$a_n=a$$ for all $$n>N$$.