Cauchy Sequence¶

Definition¶

Enrichment: non-assessable in 2026-27

Cauchy sequences are presented in Math I: Calculus as enrichment. They are not assessable in 2026-27; the exam blueprint states this explicitly for the sets-and-sequences question.

A sequence $(a_n)_{n=1}^\infty$ of real numbers is a Cauchy sequence if, for every positive real number $\varepsilon\text{,}$ there exists an $N\geq1$ such that $|a_j-a_k|<\varepsilon$ for all $j,k\geq N\text{.}$

Lecture 2 presents the same quantifier pattern first for rational sequences and positive rational $\varepsilon\text{;}$ Lecture 3 states the real-sequence version above. In either version, the essential point is that one choice of $N$ must work for every pair $j,k\geq N\text{.}$

Interpretation¶

A Cauchy sequence is one whose terms eventually become arbitrarily close to each other. The definition compares terms only with each other, never with a candidate limit, so it can be checked when the limit is unknown.

Compare it with the definition of a limit: convergence to $L$ asks for $|a_n-L|<\varepsilon$ $ε$ , the Cauchy condition for $|a_j-a_k|<\varepsilon$ $ε$ , where the absolute value $|a_j-a_k|$ is the distance between two terms. A convergent sequence is always Cauchy: if all terms from $N$ on lie within $\varepsilon/2$ of $L$ , then by the triangle inequality any two of them lie within $\varepsilon$ of each other. The converse is the Cauchy criterion of the Key Results, and it needs the real numbers: the decimal truncations $1,\ 1.4,\ 1.41,\ 1.414,\ldots$ of $\sqrt{2}$ form a Cauchy sequence of rational numbers with no limit in $\mathbb{Q}$ .

The inequality in the definition bounds all pairs in the same tail; checking only consecutive differences is insufficient. For $a_n=\sqrt{n}$ $n$ , the differences $a_{n+1}-a_n=\frac{1}{\sqrt{n+1}+\sqrt{n}}$ tend to $0$ , yet $a_{4n}-a_n=\sqrt{n}\ge1$ for every $n\ge1$ $1$ , so the sequence is not Cauchy.

In Math III the same definition is used with more general distances than $|a_j-a_k|$ $a_{k} ∣$ : see Cauchy sequence in a metric space.

Key Results¶

Cauchy criterion. A sequence of real numbers is convergent if and only if it is a Cauchy sequence. This relies on the completeness of $\mathbb{R}$ (every nonempty set that is bounded above has a supremum) and fails in $\mathbb{Q}\text{;}$ the course uses it as a criterion and does not construct $\mathbb{R}$ from Cauchy sequences.
The quantifier order matters: an arbitrary $\varepsilon>0$ is given first, and only then is $N$ chosen so that all $j,k\geq N$ work.

Examples¶

$a_n=(-1)^n/n$ is a Cauchy sequence.
$b_n=(-1)^n$ is not a Cauchy sequence.
$a_n=1/n$ is a Cauchy sequence: given $\varepsilon>0\text{,}$ any integer $N>1/\varepsilon$ works.

The terms of $a_n=\sin n/n$ crowd together as $n$ grows (a Cauchy sequence); those of $b_n=\sin n$ keep jumping across $[-1,1]$ (not Cauchy).

Prerequisites¶

Sources¶

Math I: Calculus · Lectures 2–3 (enrichment, non-assessable)

Aliases: fundamental sequence Symbols: epsilon, infinity, rational numbers, real numbers, square root

Last updated: 20 October 2026 03:02