# Cauchy Sequence with Finite Elements Prepended is Cauchy Sequence

## Theorem

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

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

Let $N \in \N$

Let $\sequence {y_n}$ be the sequence defined by:

- $\forall n, y_n = x_{N+n}$

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

Then:

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

## Proof

Given $\epsilon \gt 0$

By the definition of a Cauchy sequence then:

- $\exists N': \forall n, m \gt N', \norm {y_n - y_m} \lt \epsilon$

Hence $\forall n, m \gt \paren {N'+N}$:

\(\displaystyle \norm {x_n - x_m }\) | \(=\) | \(\displaystyle \norm {y_{n-N } - y_{m-N } }\) | $\quad$ $n, m \gt N$ | $\quad$ | |||||||||

\(\displaystyle \) | \(\lt\) | \(\displaystyle \epsilon\) | $\quad$ $n-N, m-N \gt N'$ | $\quad$ |

The result follows.

$\blacksquare$