Convergent Sequence with Finite Elements Prepended is Convergent Sequence

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:

$\forall n \in \N: y_n = x_{N + n}$

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 $\epsilon \in \R_{>0}$ be given.

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

$\exists N' \in \R_{>0}: \forall n \in \N: n > N' \implies \norm {y_n - l} < \epsilon$

Hence:

\(\ds \forall n > \paren {N' + N}: \, \)

\(\ds \norm {x_n - l}\)

\(=\)

\(\ds \norm {y_{n - N} - l}\)

$n > N$

\(\ds \)

\(<\)

\(\ds \epsilon\)

$n - N > N'$

The result follows.

$\blacksquare$