ANALYSIS 2026-05-18

Cauchy Criterion for Convergence

Theorem

One of the fundamental obstacles in analysis is proving that a sequence converges when the limit is not known in advance. The Cauchy criterion resolves this elegantly by characterising convergence entirely in terms of the internal behaviour of the sequence.

Definition

(Cauchy Sequence)

A sequence $\{a_n\}$ in $\mathbb{R}$ is called a Cauchy sequence if

\forall\,\varepsilon>0\;\exists\,N\in\mathbb{N}\;\forall\,m,n>N:\quad|a_n-a_m|<\varepsilon.

Intuitively, the terms become arbitrarily close to each other as the index grows.

Theorem

(Cauchy's Convergence Criterion)

A sequence $\{a_n\}$ of real numbers converges if and only if it is a Cauchy sequence.

The forward direction ( $\Rightarrow$ ) is straightforward. If $a_n\to L$ , then for any $\varepsilon>0$ we can find $N$ such that $|a_n-L|<\varepsilon/2$ for all $n>N$ , and the triangle inequality gives the Cauchy condition immediately. The non-trivial direction uses the completeness of $\mathbb{R}$ .

Lemma

(Cauchy Sequences are Bounded)

Every Cauchy sequence $\{a_n\}$ is bounded. Indeed, taking $\varepsilon=1$ , choose $N$ so that $|a_n-a_m|<1$ for all $m,n>N$ . Then $|a_n|\le|a_{N+1}|+1$ for $n>N$ , hence

M:=\max\bigl\{|a_1|,\ldots,|a_N|,|a_{N+1}|+1\bigr\}

satisfies $|a_n|\le M$ for all $n\in\mathbb{N}$ .

Proof

(Proof of the Cauchy Criterion ($\Leftarrow$))

(1)Let $\{a_n\}$ be Cauchy. By the lemma it is bounded.
(2)By the Bolzano–Weierstrass Theorem, every bounded sequence has a convergent subsequence; pick $\{a_{n_k}\}\to L$ .
(3)Let $\varepsilon>0$ . Since $\{a_n\}$ is Cauchy, $\exists\,N_1$ such that $|a_n-a_m|<\varepsilon/2$ for all $m,n>N_1$ .
(4)Since $a_{n_k}\to L$ , $\exists\,K$ such that $|a_{n_k}-L|<\varepsilon/2$ for all $k>K$ . Pick $k$ large enough that $n_k>N_1$ and $k>K$ .
(5)For any $n>N_1$ : $|a_n-L|\le|a_n-a_{n_k}|+|a_{n_k}-L|<\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon.$
(6)Since $\varepsilon$ was arbitrary, $a_n\to L$ . $\qquad$

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Corollary

(Completeness of $\mathbb{R}$)

The Cauchy criterion holds in $\mathbb{R}$ precisely because $\mathbb{R}$ is complete: every Cauchy sequence converges. In $\mathbb{Q}$ this fails — the sequence of rational approximations to $\sqrt{2}$ is Cauchy but has no rational limit.

Remark

(Generalisation to Metric Spaces)

A metric space $(X,d)$ in which every Cauchy sequence converges is called a complete metric space. Examples include $\mathbb{R}^n$ , all Hilbert spaces, and the $p$ -adic numbers $\mathbb{Q}_p$ . The criterion generalises verbatim by replacing $|a_n-a_m|$ with $d(a_n,a_m)$ .