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 $\forall n, m > \paren {N' + N}$:
\(\ds \norm {x_n - x_m }\) | \(=\) | \(\ds \norm {y_{n - N} - y_{m - N} }\) | $n, m > N$ | |||||||||||
\(\ds \) | \(<\) | \(\ds \epsilon\) | $n - N, m - N > N'$ |
The result follows.
$\blacksquare$