# Modulus of Cauchy Sequence has Limit

## Theorem

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

Let $\sequence {x_n}$ be a Cauchy sequence in $R$.

Then $\sequence {\norm {x_n}}$ has a limit in $\R$.

That is,

$\exists l \in \R: \displaystyle \lim_{n \to \infty} {\norm {x_n}} = l$

## Proof

By Modulus of Cauchy Sequence is Cauchy Sequence then $\sequence {\norm {x_n}}$ is a real Cauchy sequence.

By Real Sequence is Cauchy iff Convergent the sequence $\sequence {\norm {x_n}}$ has a limit in $\R$.

$\blacksquare$