Theorem
Let
be the set of Cauchy sequences in a metric space
The relation
on defined by
is an equivalence relation.
Proof
since
since
by the metric property.
since
Henceis an equivalence relation on the set of Cauchy sequences.
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Theorem
Letbe the set of Cauchy sequences in a metric spaceThe relationon defined byis an equivalence relation.
Proof
since
sinceby the metric property.
since
Henceis an equivalence relation on the set of Cauchy sequences.