1 Cauchy Sequences
Cauchy (mutual) convergence is a useful result when you have to prove convergence but do not have a good candidate for a limit. Cauchy convergence requires only information on pairs \((x_{n},x_{m})\).
Informally, a real sequence \(\{ x_{n} \}\) is Cauchy when the terms of the sequence become arbitrarily close together as \(n\) gets very large. We write a more concrete metric based definition of Cauchy sequences using the standard Euclidean metric.
A real sequence \(\{ x_{n} \}\) is Cauchy if \[\forall \epsilon >0, \exists N=N_{\epsilon}~s.t.~|x_{n}-x_{m}|>\epsilon ,\forall n,m>N,\]or in other words \(x_{n}-x_{m}\to 0\) as \(n,m \to \infty\) independently.
This leads to the following important result in Metric Spaces and Functional Analysis in general known as the Cauchy characterization of convergence for sequences.
A real sequence \(\{ x_{n} \}\) is convergent in the reals if and only if it is Cauchy.
PROOF: