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Let X be a metric space. (a) Call two Cauchy sequences {pn},{qn} in X equivalent if limn →∞ d(pn, qn)=0 Prove that this is an equivalence relation. (b) Let X^* be the set of all equivalence classes so obtained. If P ∈X^*, Q ∈X^*, {pn} ∈P,{qn} ∈Q, define Δ(P, Q)=limn →∞ d(pn, qn) by Exercise 23, this limit exists. Show that the number Δ(P, Q) is unchanged if {pn} and {q*} are replaced by equivalent sequences, and hence that Δis a distance function in X^*. (c) Prove that the resulting metric space
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