Let $(a_n)_n$ be a convergent sequence of integers , what can we say about $(a_n)_n$

Let $(a_n)_n$ be a convergent sequence of integers , what can we say about $(a_n)_n$?

(I don't understand what is meant by this question)

Solution 1:

HINT: Suppose that the sequence converges to some number $L$. Then for each $\epsilon>0$ there is an $m_\epsilon\in\Bbb N$ such that $|a_n-L|<\epsilon$ whenever $n\ge m_\epsilon$. In particular, there is an $m_{1/4}\in\Bbb N$ such that $$|a_n-L|<\frac14$$ whenever $n\ge m_{1/4}$. Using the triangle inequality, we see that if $k,n\ge m_{1/4}$, then

$$|a_k-a_n|\le|a_k-L|+|L-a_n|<\frac14+\frac14=\frac12\;.$$

Now remember: each of these terms $a_n$ is an integer. What can you say about integers that are less than $\frac12$ apart?