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 \((x_{n},x_{m})\).
A real sequence \(\{ x_{n} \}\) converges if and only if it is Cauchy, that is \[\forall \epsilon >0, \exists N=N_{\epsilon}~s.t.~|x_{n}-x_{m}|>\epsilon ,\forall n,m>N,\]or in other words \(x_{n}-x_{m}\to 0\) as \(n,m \to \infty\) independently.