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 $\epsilon > 0$:

By the definition of a Cauchy sequence then:

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

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

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

The result follows.

$\blacksquare$


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