In mathematics, a Cauchy sequence captures the idea that the terms of a sequence eventually cluster together even if their common limit is not known or does not lie in the ambient space. Informally, the farther one goes out in the sequence the closer any two later terms are to one another. This intrinsic notion of convergence is independent of a specific limit point and is fundamental in analysis.

Definition

In a metric space (X,d), a sequence (x_n) is called Cauchy if for every ε>0 there exists N such that for all m,n ≥ N one has d(x_m,x_n) < ε. Equivalently, the pairwise distances between terms tend to zero as indices go to infinity. The definition extends to uniform spaces by replacing distances with entourages.

Key properties

  • Every convergent sequence is Cauchy, because terms approach the common limit and thus each other.
  • The converse holds in complete spaces: every Cauchy sequence converges to a point of the space.
  • Cauchy behaviour depends on the ambient space; a sequence can be Cauchy in one space and divergent in a subspace that is not complete.

Examples and importance

A classical example uses rational numbers: decimal approximations of √2 form a Cauchy sequence in the rationals that does not converge within the rationals, illustrating that Q is not complete. Cauchy sequences underpin the construction of completions (for example completing Q to obtain the real numbers) and appear throughout real analysis, functional analysis, and numerical methods where internal convergence criteria are needed.

The concept is named for Augustin-Louis Cauchy, who contributed to rigorous analysis in the 19th century. Variants include Cauchy nets and Cauchy filters used in general topology. Recognizing whether a space is complete often reduces to checking that every Cauchy sequence converges, making the notion a practical tool for studying metric and uniform spaces.