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

Let $\epsilon > 0$ be given.

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

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

Hence:

\(\, \displaystyle \forall n > \paren {N' + N}: \, \) \(\displaystyle \norm {x_n - l}\) \(=\) \(\displaystyle \norm {y_{n - N} - l}\) $n > N$
\(\displaystyle \) \(<\) \(\displaystyle \epsilon\) $n - N > N'$

The result follows.

$\blacksquare$


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