## 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