Convergent Sequence with Finite Elements Prepended is Convergent Sequence
Jump to navigation
Jump to search
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.
$\blacksquare$