Proof That the Metric on the Quotient Space of Cauchy Sequences is Well Defined

Theorem

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

Supposethen

Set

From the triangle inequality

LetThere existssuch that such thatsuch that

Takethenand

Since

Similarly,

Hence