Cauchy sequence

Updated 2026-07-04

A Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. Formally, in a metric space $(X,d)$ a sequence $(x_n)_{n\in\mathbb N}$ is called Cauchy if for every $\varepsilon>0$ there exists an integer $N$ such that

$$ \forall m,n\ge N,\qquad d(x_m,x_n)<\varepsilon . $$

The definition can be generalized to uniform spaces by replacing the metric $d$ with a uniform structure; in that setting a sequence is Cauchy if for every entourage $U$ there is $N$ with $(x_m,x_n)\in U$ for all $m,n\ge N$.

Key Properties

Convergence implies Cauchy. In any metric (or uniform) space, if $(x_n)$ converges to a limit $x$, then it is Cauchy.
Completeness. A metric space is called complete if every Cauchy sequence in the space converges to a limit that lies in the space. The real numbers $\mathbb{R}$ with the usual absolute‑value metric are a classic example of a complete space; the rational numbers $\mathbb{Q}$ are not, because there exist Cauchy sequences of rationals that converge to irrational limits.
Uniqueness of limit. In a Hausdorff (in particular, metric) space, a Cauchy sequence can have at most one limit.
Cauchy subsequences. Any subsequence of a Cauchy sequence is itself Cauchy. Conversely, a sequence that possesses a Cauchy subsequence need not be Cauchy in general.

Examples

Real numbers. The sequence defined by $x_n = 1/n$ in $\mathbb{R}$ is Cauchy because $|x_m - x_n| \le 1/N$ for all $m,n\ge N$.
Rational approximations of $\sqrt{2}$. The decimal truncations $1, 1.4, 1.41, 1.414, \dots$ form a Cauchy sequence in $\mathbb{Q}$ that does not converge in $\mathbb{Q}$ but does converge to $\sqrt{2}$ in $\mathbb{R}$.
Sequence in a discrete metric space. In a space where $d(x,y)=1$ for $x eq y$ and $0$ otherwise, a sequence is Cauchy only if it is eventually constant.

Historical Note

The concept is named after the French mathematician Augustin-Louis Cauchy (1789–1857). Cauchy introduced the idea in his work on the foundations of analysis, particularly in the context of series and the rigorous definition of limits. The modern formal definition of Cauchy sequences as a tool for describing completeness was crystallized in the 19th century by mathematicians such as Karl Weierstrass and Georg Cantor.

Relation to Other Concepts

Cauchy net/ filter. In general topological spaces, sequences may be insufficient to capture convergence behavior. Cauchy nets and Cauchy filters extend the notion of Cauchy sequences to arbitrary topological spaces equipped with a uniform structure.
Banach spaces. A Banach space is a complete normed vector space; completeness is expressed in terms of Cauchy sequences with respect to the norm‑induced metric.
Metric completions. Every metric space $(X,d)$ admits a completion: a complete metric space $(\widehat{X},\widehat{d})$ containing an isometric copy of $X$ such that each Cauchy sequence in $X$ converges in $\widehat{X}$.

References

R. L. Moore, Foundations of Point Set Theory, 1965.
J. M. Lee, Introduction to Topological Manifolds, 2nd ed., 2019.
G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed., 1999.