# Cauchy Sequence is Bounded

## 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.