Definition:Convergent Sequence/Normed Division Ring

Definition

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.

Let $\sequence {x_n}$ be a sequence in $R$.

Definition 1

The sequence $\sequence {x_n}$ converges to $x \in R$ in the norm $\norm {\, \cdot \,}$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \norm {x_n - x} < \epsilon$

Definition 2

The sequence $\sequence {x_n}$ converges to $x \in R$ in the norm $\norm {\, \cdot \,}$ if and only if:

$\sequence {x_n}$ converges to $x$ in the metric induced by the norm $\norm {\, \cdot \,}$

Definition 3

The sequence $\sequence {x_n}$ converges to $x \in R$ in the norm $\norm {\, \cdot \,}$ if and only if:

the real sequence $\sequence {\norm {x_n - x} }$ converges to $0$ in the reals $\R$

Limit of Sequence

Let $\sequence {x_n}$ converge to $x \in R$.

Then $x$ is a limit of $\sequence {x_n}$ as $n$ tends to infinity which is usually written:

$\displaystyle x = \lim_{n \mathop \to \infty} x_n$