Proof That the Metric on the Quotient Space of Cauchy Sequences is Well Defined
Letbe the set of all Cauchy sequences on a metric space
Ifis a Cauchy sequence inthenis the equivalence class containingandis the quotient space.
The metric on the quotient space, defined asis well defined, so that
From the triangle inequality
LetThere existssuch that such thatsuch that