Cauchy Sequence is Bounded
Jump to navigation
Jump to search
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.