# Modulus of Cauchy Sequence is Cauchy Sequence

This article has been proposed for deletion. In particular: This theorem has been replaced by Norm Sequence of Cauchy Sequence has Limit |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

- $\sequence {\norm {x_n}}$ is a real Cauchy sequence in $\R$.

## Proof

Given $\epsilon \gt 0$.

By the definition of Cauchy sequence then:

- $\exists N \in \N: \forall n, m > N, \norm {x_n - x_m} \lt \epsilon$

By the reverse triangle inequality, then:

- $\forall n, m \gt N: \cmod {\norm {x_n} - \norm {x_m}} \le \norm {x_n - x_m} \lt \epsilon$

From the definition of a real Cauchy sequence the result follows.

$\blacksquare$