Cauchy Sequence in a Metric Space¶

Definition¶

A sequence of points $(x_i)_{i=1}^\infty$ in a metric space $(X,d)$ is called a Cauchy sequence if, for every $\varepsilon > 0$ $0$ , there exists an integer $N\geq 1$ such that the distance $d(x_j,x_k)<\varepsilon$ for all $j, k \geq N$ $N$ .

Interpretation¶

A Cauchy sequence is one whose terms eventually all lie close to each other. It is the same idea as a Cauchy sequence of real numbers, but with the absolute value replaced by a general metric. Unlike the definition of convergence, it never mentions a limit, so it can be checked using the terms alone.

Points of a sequence spiralling inwards inside a rectangle representing the space; from some index on all terms lie within a small circle, close to each other.

From some index $N$ on, all terms in the figure lie inside the small dashed circle, close enough that $d(x_j,x_k)<\varepsilon$ for all $j,k\ge N$ $N$ . The terms get closer to each other, but a Cauchy sequence need not converge to a point of $X$ unless the space is complete.

Raise $N$ until all tail points lie within $\varepsilon$ of each other, without using the limit at all.

The condition concerns all pairs $j,k\ge N$ $N$ , not only consecutive terms. The partial sums $x_k=1+\tfrac{1}{2}+\cdots+\tfrac{1}{k}$ of the harmonic series have $|x_{k+1}-x_k|=\tfrac{1}{k+1}\to0$ $0$ , yet $x_{2k}-x_k\ge k\cdot\tfrac{1}{2k}=\tfrac{1}{2}$ for every $k$ , so they are not Cauchy.

Every convergent sequence is Cauchy, by the triangle inequality $d(x_j,x_k)\le d(x_j,x)+d(x,x_k)$ $d (x, x_{k})$ . The converse holds exactly in complete metric spaces, such as $\mathbb{R}^n$ with any $p$ -metric, and fails in $\mathbb{Q}$ , as the Examples show. Being Cauchy depends on the metric, but strongly equivalent metrics have the same Cauchy sequences, because $d_q(x_j,x_k)\le b\,d_p(x_j,x_k)$ and $d_p(x_j,x_k)\le\tfrac{1}{a}\,d_q(x_j,x_k)$ $\frac{1}{a} d_{q} (x_{j}, x_{k})$ .

Key Results¶

While all convergent sequences are Cauchy, not every Cauchy sequence converges because a sequence can cluster together (Cauchy property) around a limit that does not exist within the current space.

Examples¶

Take the rational numbers $\mathbb{Q}$ with the usual metric $d(x,y)=|x-y|$ $y ∣$ . The sequence $a_n=\left(1+\dfrac{1}{n}\right)^n$ consists of rational numbers and converges in $\mathbb{R}$ to $\mathrm{e}$ , so it is Cauchy. Since $\mathrm{e}$ is irrational, it does not converge to any point of $\mathbb{Q}$ : the space $(\mathbb{Q},d)$ is not complete.

Prerequisites¶

Metric and Metric Space

Sources¶

Math III: Analysis · Lecture 4

Symbols: epsilon, infinity, rational numbers, real numbers

Last updated: 20 October 2026 03:02