Convergent Sequence with Finite Elements Prepended is Convergent 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 convergent sequence in $R$ with limit $l$.

Then:

$\sequence {x_n}$ is a convergent sequence in $R$ with limit $l$.

Proof

Given $\epsilon \gt 0$

By the definition of a convergent sequence in $R$ with limit $l$ then:

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

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

\(\displaystyle \norm {x_n - l }\) \(=\) \(\displaystyle \norm {y_{n-N } - l }\) $\quad$ $n \gt N$ $\quad$
\(\displaystyle \) \(\lt\) \(\displaystyle \epsilon\) $\quad$ $n-N \gt N'$ $\quad$

The result follows.

$\blacksquare$


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