Convergent Sequence is Cauchy Sequence

From ProofWiki
Jump to navigation Jump to search

Theorem

Metric Space

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


Every convergent sequence in $M$ is a Cauchy sequence.


Normed Division Ring

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


Every convergent sequence in $R$ is a Cauchy sequence.


Normed Vector Space

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


Every convergent sequence in $X$ is a Cauchy sequence.