Proof That a Metric on a Metric Space Induces a Metric on the Quotient Space of Cauchy Sequences
Theorem
Ifis a metric space with metricandis the set of Cauchy sequences on withifthen
is metric on the quotient space
Proof
is well defined on
M1sinceIfthenand ifthen
M2
M3
Henceis metric on the quotient space