Theorem
Every Cauchy sequence
in a metric space
is bounded and totally bounded.
Proof
Letbe a Cauchy sequence in a metric space
A subset
of a metric spaceis totally bounded ifhas an
- net for every
(an- net foris a finite set of points
such that for every
there is
such that
).
Let
Sinceis Cauchy, there exists
such that
Hence the diameter of the set
is at most
The- net for the sequenceis then