Theorem
Suppose we have a metric space
Suppose
is a Cauchy sequence in
Define
as follows:
where
Then
is isometric to
Proof
Suppose
Then
and
Henceis isometric to
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Theorem
Suppose we have a metric spaceSupposeis a Cauchy sequence in
Defineas follows:where
Thenis isometric to
Proof
Suppose
Then
and
Henceis isometric to