Cauchy Sequence with Finite Elements Prepended is Cauchy Sequence

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

\(\displaystyle \norm {x_n - x_m }\) \(=\) \(\displaystyle \norm {y_{n-N } - y_{m-N } }\) $n, m \gt N$
\(\displaystyle \) \(\lt\) \(\displaystyle \epsilon\) $n-N, m-N \gt N'$

The result follows.

$\blacksquare$


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