Let (M, d) be a metric space and let C be the space of all Cauchy sequences in M

Question

Let (M, d) be a metric space and let C be the space of all Cauchy sequences in M. Recall that two Cauchy sequences (pn) and (qn) are said to be co-Cauchy if d(pn, qn) rightarrow 0 as n rightarrow infinity. Show that co-Cauchyness is an equivalence relation on C. Let M denote the set of equivalence classes under this relation. (In other words, we can think of each element of M as a collection of Cauchy sequences that are all co-Cauchy.)

Explanation / Answer

1)reflexive if Pn is a cauchy sequence then d(Pn,Pn)--->0 as n-->infinity 2)symmetric let Pn and Qn are two cauchy sequences let d(Pn,Qn)-->0 as n-->infinity this means that |Pn-Qn|--->0 as n---->infinity now |Pn-Qn|=|Qn-Pn| hence if |Pn-Qn|--->0 as n---->infinity it means |Qn-Pn|---->0 as n--->infinty hence d(Qn,Pn)-->0 as n----> infinity hence relation is symmetric 2)transistive let d(Pn,Qn)--->0 as n--> infinity and d(Qn,Rn)--->0 as n---> infinity now this means |Pn-Qn|--->0 as n---->infinity |Qn-Rn|--->0 as n---->infinity now |Pn-Rn|=|Pn-Qn+Qn-Rn| now by triangle inequality |Pn-Rn|=|Pn-Qn+Qn-Rn|infinity so relation is transistive

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