Cauchy Sequence is Bounded

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Theorem

Metric Space

Let $M = \struct {A, d}$ be a metric space.


Then every Cauchy sequence in $M$ is bounded.


Normed Division Ring

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring.

Every Cauchy sequence in $R$ is bounded.


Normed Vector Space

Let $V = \struct {X, \norm {\,\cdot\,} }$ be a normed vector space.


Every Cauchy sequence in $X$ is bounded.


Real Numbers

Every Cauchy sequence in $\R$ is bounded.