Cauchy Sequence

A Cauchy sequence is a sequence for which the terms eventually become arbitrarily close to each other.

Definition

In a metric space \(X\), a sequence \(\{x_{k}\}_{k \in \mathbb{N}}\) is a Cauchy sequence if and only if for any \(\epsilon > 0\) there exists an \(N \in \mathbb{N}\) such that:

\[ m, n > N \implies d(x_{n}, x_{m}) < \epsilon.\]

It is important to note that points getting pairwise close together is not sufficient for it to be Cauchy. One example is the sequence of partial sums of the harmonic series

\[ x_k = \sum_{n=1}^{k} = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{k}\]

where \(x_{k + 1} - x_k = \frac{1}{k}\) which approaches zero, yet the series is unbounded and thus cannot be Cauchy.

The main utility of this idea is for the notion of completeness. That is, a way of expressing the notion of sequences which "should" converge, but the space is missing the points necessary for that convergence to actually hold.