Cauchy Sequence

Definition §

A Cauchy sequence (fundamental sequence) is a sequence whose elements get arbitrarily close together as it progresses. Given any arbitrarily small distance, all but a finite number of elements of the sequence are within that distance of each other. More formally, a sequence $\{a_n{\}}_{n=1}^{\infty }$ is Cauchy if for any arbitrary $\epsilon >0$, there exists a sufficiently large $N\in {\mathbb{Z}}^{+}$ where $\forall p,q>N$, $|a_p-a_q|<\epsilon$.