Definition:Cauchy Sequence

Definition

Metric Space

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

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

Then $\left \langle {x_n} \right \rangle$ is a Cauchy sequence iff:

$\forall \epsilon \in \R: \epsilon > 0: \exists N: \forall m, n \in \N: m, n \ge N: d \left({x_n, x_m}\right) < \epsilon$

Real Numbers

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

Then $\left \langle {x_n} \right \rangle$ is a Cauchy sequence iff:

$\forall \epsilon \in \R: \epsilon > 0: \exists N: \forall m, n \in \N: m, n \ge N: \left|{x_n - x_m}\right| < \epsilon$

Considering the real number line as a metric space, it is clear that this is a special case of the definition for a metric space.

Rational Numbers

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

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

Then $\left \langle {x_n} \right \rangle$ is a Cauchy sequence iff:

$\forall \epsilon \in \Q: \epsilon > 0: \exists N: \forall m, n \in \N: m, n \ge N: \left|{x_n - x_m}\right| < \epsilon$

Considering the set of rational numbers as a metric space, it is clear that this is a special case of the definition for a metric space.

Cauchy Criterion

That is, 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.

This condition is known as the Cauchy criterion.

Also see

• A Convergent Sequence is Cauchy Sequence.

• A complete metric space is defined as being a metric space in which the converse holds, i.e. a Cauchy sequence is convergent.

• The space $\R$ of real numbers is a complete metric space.

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

Source of Name

This entry was named for Augustin Louis Cauchy.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 5$: Complete Metric Spaces
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Definition $1.2.8$
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 5.16$