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A metric space metric space \((X,d)\) is `Cauchy complete' if every Cauchy sequence in \(X\) has a limit in \(X\). For example: \(\mathbb{R}\) is Cauchy complete by definition, while \(\mathbb{Q}\) is not, as the sequence \(\sum_{r=0}^n\frac{1}{n!}\) is Cauchy but its limit, \(e\), is not in \(\mathbb{Q}\). Given a metric space \((X,d)\), its Cauchy (or metric) completion \((\bar{X},\bar{d})\) is the quotient of the set of all Cauchy sequences in \(X\) modulo the equivalence relation \((a_i)_{i=0}^\infty\sim(b_i)_{i=0}^\infty:\Leftrightarrow\lim_{i\to\infty}d(a_i,b_i)=0\), so that two sequences are equivalent if they share a limit point, with an induced metric \(\bar{d}\left(\left[(a_i)_{i=0}^\infty\right],\left[(b_i)_{i=0}^\infty\right]\right):=\lim_{i\to\infty}d(a_i,b_i)\). \(\mathbb{R}\) is the metric completion of \(\mathbb{Q}\).