# Convergent Sequence is Cauchy Sequence

## 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.