Definition:Limit of Sequence/Normed Division Ring

Definition

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

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

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:

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

Also known as

A limit of $\sequence {x_n}$ as $n$ tends to infinity can also be presented more tersely as a limit of $\sequence {x_n}$ or even just limit of $x_n$.

Some sources present $\ds \lim_{n \mathop \to \infty} x_n$ as $\lim_n x_n$.

Also see

• Results about limits of sequences can be found here.

Sources

• 2007: Svetlana Katok: p-adic Analysis Compared with Real: $\S 1.2$: Normed Fields, Definition $1.7$