Definition:Limit of Sequence (Number Field)

Definition

Real Numbers

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

Let $\sequence {x_n}$ converge to a value $l \in \R$.

Then $l$ is a limit of $\sequence {x_n}$ as $n$ tends to infinity.

Rational Numbers

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

Let $\sequence {x_n}$ converge to a value $l \in \R$, where $\R$ denotes the set of real numbers.

Then $l$ is a limit of $\sequence {x_n}$ as $n$ tends to infinity.

Complex Numbers

Let $\sequence {z_n}$ be a sequence in $\C$.

Let $\sequence {z_n}$ converge to a value $l \in \C$.

Then $l$ is a limit of $\sequence {z_n}$ as $n$ tends to infinity.

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 $\lim_n x_n$.