CauchySequence

A Cauchy sequence is a sequence $(a_n)$ of real numbers that satisfies the following condition: For every positive real number $\epsilon > 0$, there exists a positive integer $N_\epsilon$ such that for all integers $m, n \geq N_\epsilon$, the absolute difference between $a_m$ and $a_n$ is less than $\epsilon$; that is

$$\text{ there exists an } N_\epsilon \in \mathbb{N} \text{ such that } |a_m - a_n| < \epsilon \forall m, n \geq N_\epsilon \forall \epsilon > 0$$

In other words, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses.