\begin{definition}[Definition:Cauchy Sequence/Cauchy Criterion]
The '''Cauchy criterion''' is the condition:
:For any (strictly) positive real number $\epsilon \in \R_{>0}$, for a sufficiently large natural number $N \in \N$, the difference between the $m$th and $n$th terms of a Cauchy sequence, where $m, n \ge N$, will be less than $\epsilon$.
Informally:
:For any number you care to pick (however small), if you go out far enough into the sequence, past a certain point, the difference between any two terms in the sequence is less than the number you picked.
Or to put it another way, the terms get arbitrarily close together the farther out you go.
\end{definition}