Cauchy Sequence

A Sequence $(a_{n})$ is called a Cauchy Sequence if, $\forall ϵ > 0$ there $\exists N \in N$ where any $m, n \geq N$ follows that:

| a_{n} - a_{m} | < ϵ

Consider the definition of convergence:

Convergence of a sequence

A Sequence $(a_{n})$ of real numbers converges to some $L \in R$ if:
$\forall ε > 0 \exists N \in N \forall n \geq N (| a_{n} - L | < ε)$

The definition of convergence asserts that, given an arbitrary positive $ϵ$ we can find a point in the sequence $N$ such that all points after $n \geq N$ are closer to the limit $a$ than the given $ϵ$ .

On the other hand, a sequence is Cauchy if, for every $ϵ$ there is a point in the sequence after which the terms are all closer to each other than the given $ϵ$ .

We'll argue that the two definitions are equivalent: convergent sequences are cauchy, and a cauchy sequence is convergent. The significance of this definition is that there is no mention of the value of the limit, similar to the Monotone Convergence Theorem. See Cauchy Sequences Converge.