Cauchy sequence

Definition

Let $(a_n)$ be a real sequence. The sequence is a Cauchy sequence if the following holds for each real number $\varepsilon > 0$. There is a natural number $N$ such that $|a_m - a_n| < \varepsilon$ for all natural numbers $m, n \ge N$.

Remarks

In a metric space $(X, d)$, the definition is the same with $d(a_m, a_n)$ in place of $|a_m - a_n|$. The definition does not mention a limit. In $\mathbb{R}$, the Cauchy criterion shows that the Cauchy sequences are exactly the convergent sequences. It is not enough that $|a_{n+1} - a_n| \to 0$. The sequence $a_n = \sqrt{n}$ has this property, but it is not Cauchy.

Explanations

L1 human

In a Cauchy sequence, the terms become close to each other, not only to their neighbours. From some index on, any two terms differ by less than the error that you allow.