# Definition:Cauchy Sequence

## Definition

### Metric Space

Let $M = \struct {A, d}$ be a metric space.

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

Then $\sequence {x_n}$ is a **Cauchy sequence** if and only if:

- $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \map d {x_n, x_m} < \epsilon$

### Normed Vector Space

Let $\struct {V, \norm {\,\cdot\,} }$ be a normed vector space.

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

Then $\sequence {x_n}$ is a **Cauchy sequence** if and only if:

- $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \norm {x_n - x_m} < \epsilon$

### Normed Division Ring

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

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

Then $\sequence {x_n} $ is a **Cauchy sequence in the norm $\norm {\, \cdot \,}$** if and only if:

- $\sequence {x_n}$ is a cauchy sequence in the metric induced by the norm $\norm {\, \cdot \,}$

That is:

- $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \norm {x_n - x_m} < \epsilon$

### Topological Vector Space

Let $\struct {X, \tau}$ be a topological vector space.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$.

We say that $\sequence {x_n}_{n \mathop \in \N}$ is **Cauchy** if and only if:

- for each open neighborhood $V$ of ${\mathbf 0}_X$ there exists $N \in \N$ such that:

- $x_n - x_m \in V$ for each $n, m \ge N$.

## Standard Number Fields

### Complex Numbers

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

Then $\sequence {z_n}$ is a **Cauchy sequence** if and only if:

- $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \size {z_n - z_m} < \epsilon$

where $\size {z_n - z_m}$ denotes the complex modulus of $z_n - z_m$.

### Real Numbers

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

Then $\sequence {x_n}$ is a **Cauchy sequence** if and only if:

- $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \size {x_n - x_m} < \epsilon$

### Rational Numbers

The concept can also be defined for the set of rational numbers $\Q$:

Let $\sequence {x_n}$ be a rational sequence.

Then $\sequence {x_n}$ is a **Cauchy sequence** if and only if:

- $\forall \epsilon \in \Q_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \size {x_n - x_m} < \epsilon$

where $\Q_{>0}$ denotes the set of all strictly positive rational numbers.

## Cauchy Criterion

The **Cauchy criterion** is the condition:

- For any (strictly) positive real number $\epsilon \in \R_{>0}$, for a sufficiently large natural number $N \in \N$, the difference between the $m$th and $n$th terms of a Cauchy sequence, where $m, n \ge N$, will be less than $\epsilon$.

Informally:

- For any number you care to pick (however small), if you go out far enough into the sequence, past a certain point, the difference between any two terms in the sequence is less than the number you picked.

Or to put it another way, the terms get arbitrarily close together the farther out you go.

## Also known as

A **Cauchy sequence** is also known as a **fundamental sequence**.

## Also see

- Definition:Complete Metric Space: a metric space in which the converse holds, that is a Cauchy sequence is convergent.

Thus in $\R$ and $\C$ a **Cauchy sequence** and a convergent sequence are equivalent concepts.

- Results about
**Cauchy sequences**can be found**here**.

## Source of Name

This entry was named for Augustin Louis Cauchy.

## Sources

- 1953: Walter Rudin:
*Principles of Mathematical Analysis*... (next): $3.8$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**Cauchy sequence**or**fundamental sequence**