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Cauchy Sequence**

Given a metric space \((X,d)\), a sequence \((a_i)_{i=0}^\infty\) is 'Cauchy' if the diameter of the partial sequence \((a_i)_{i=n}^\infty\) tends to 0 as \(n\to\infty\), i.e. \(\forall \epsilon>0:\exists N\geq0:\forall n,m\geq N:d(a_n,a_m)<\epsilon\) - for any given radius, there exists a point in the sequence beyond which all points are within that radius of each other.