Suppose that $\lim_{ n \to \infty }a_{n}=b\in \mathbb{R}$. Since all the number $a_{n}$ are eventually near $b$, they are all near each other! Thus, every convergent sequence obeys a `Cauchy condition` given by $ \forall \:{}\epsilon>0,\:{}\exists N\in \mathbb{N}\text{ such that if }n,k\geq{}N\text{ then }\lvert a_{n}-a_{k} \rvert <\epsilon. $ So for each $\epsilon>0$ we can find some index, $N$, where any two terms $a_{n}$ and $a_{k}$ with $n,k\geq{}N$ are less than $\epsilon$ apart. #TODO Converges on real sequence implies cauchy (Exercise 20.)