# Definition:Limit of Sequence

## Contents

## Definition

### Topological Space

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $A \subseteq S$.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $A$.

Let $\left \langle {x_n} \right \rangle$ converge to a value $\alpha \in A$.

Then $\alpha$ is known as a **limit (point) of $\left \langle {x_n} \right \rangle$ (as $n$ tends to infinity)**.

### Metric Space

Let $M = \left({A, d}\right)$ be a metric space or pseudometric space.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $M$.

Let $\left \langle {x_n} \right \rangle$ converge to a value $l \in A$.

Then $l$ is a **limit of $\left \langle {x_n} \right \rangle$ as $n$ tends to infinity**.

If $M$ is a metric space, this is usually written:

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

### Normed Division Ring

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 a value $l \in R$.

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

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

## Standard Number Fields

As:

- The set of rational numbers $\Q$ under the usual metric forms a metric space
- The real number line $\R$ under the usual metric forms a metric space
- The complex plane $\C$ under the usual metric forms a metric space

the definition of the limit of a sequence in a metric space holds for sequences in the standard number fields $\Q$, $\R$ and $\C$:

### Real Numbers

Let $\left \langle {x_n} \right \rangle$ be a sequence in $\R$.

Let $\left \langle {x_n} \right \rangle$ converge to a value $l \in \R$.

Then $l$ is a **limit of $\left \langle {x_n} \right \rangle$ as $n$ tends to infinity**.

### Rational Numbers

Let $\left \langle {x_n} \right \rangle$ be a sequence in $\Q$.

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

Then $l$ is a **limit of $\left \langle {x_n} \right \rangle$ 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 see

- Definition:Limit of Sets for an extension of this concept into the field of measure theory.