Dear Stackexchange Community,
I would like to ask how to show that a Cauchy sequence that takes on integral values is ultimately constant.
Being unfamiliar with this, do 'integral' values refer to integer values?
My solution currently is:
Let the sequence $(x_n: n \in \mathbb N)$ be Cauchy.
Then by definition, $\forall \varepsilon > 0, \exists K \in \mathbb N$ such that $\forall n, m > K, |x_n-x_m| < \varepsilon$.
I would assume to prove the statement by contradiction. Assume otherwise, that the sequence takes on more than one value (and thus not constant). However I am finding it tough to establish a contradiction.