Given a set S, let S* be the set of all Cauchy sequences. Is it true that S* is a complete metric space?
Suppose that $A, B \in S^*$. Then the metric is $\rho(A,B) = \lim_{v \to \infty} \rho(a_{v},b_{v})$.
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In what metric? – msteve Oct 19 '14 at 17:16
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Suppose that $A, B \in S^*$. Then the metric is $\rho(A,B) = \lim_{v \to \infty} \rho(a_{v},b_{v})$. – darkgbm Oct 19 '14 at 17:34
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The proposed metric is not a metric, because it can assign "distance" 0 to a pair of distinct sequences. – Greg Martin Oct 19 '14 at 17:51
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It allows S* to be a pseudometric, we can define S** to be a set consisting of equivalence classes of metrics so that $A\asymp B$ if $\rho (A,B) = 0$. Then S** will be a metric. – darkgbm Oct 19 '14 at 17:55