Cauchy Sequences

A Cauchy sequence is one whose terms become arbitrarily close to each other as the sequence progresses. In complete metric spaces like $\mathbb{R}$ , Cauchy sequences are precisely the convergent sequences — this equivalence is the Cauchy completeness criterion. Cauchy's definition (1821) provides an intrinsic characterization of convergence without reference to the limit.

Definition

Definition2.1Cauchy sequence

A sequence $(a_n)$ in $\mathbb{R}$ is Cauchy if for every $\epsilon > 0$ , there exists $N \in \mathbb{N}$ such that

$|a_m - a_n| < \epsilon \quad \text{for all } m, n \geq N.$

RemarkIntuition

A Cauchy sequence has terms that cluster together: for large enough $m$ and $n$ , $a_m$ and $a_n$ are within $\epsilon$ of each other. Unlike convergence (which requires knowledge of the limit), the Cauchy condition is checkable using only the sequence terms.

Example1/n is Cauchy

Let $a_n = 1/n$ . For $m, n \geq N$ ,

$\left|\frac{1}{m} - \frac{1}{n}\right| \leq \frac{1}{m} + \frac{1}{n} \leq \frac{2}{N}.$

Given $\epsilon > 0$ , choose $N$ such that $2/N < \epsilon$ . Then $|a_m - a_n| < \epsilon$ for all $m, n \geq N$ . So $(a_n)$ is Cauchy.

ExamplePartial sums of harmonic series are not Cauchy

Let $s_n = 1 + 1/2 + 1/3 + \cdots + 1/n$ . Then $(s_n)$ is not Cauchy. For $m = 2n$ and $n$ ,

$|s_{2n} - s_n| = \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n} \geq n \cdot \frac{1}{2n} = \frac{1}{2}.$

So $|s_m - s_n|$ does not go to zero, and $(s_n)$ is not Cauchy.

Relationship with convergence

Theorem2.1Cauchy Criterion for Convergence

A sequence $(a_n)$ in $\mathbb{R}$ converges if and only if it is Cauchy.

RemarkProof sketch

( $\Rightarrow$ ) If $a_n \to L$ , then for large $m, n$ , both $a_m$ and $a_n$ are close to $L$ , hence close to each other.

( $\Leftarrow$ ) If $(a_n)$ is Cauchy, it is bounded (proof: choose $N$ such that $|a_m - a_n| < 1$ for $m, n \geq N$ ; then all terms lie in a bounded interval). By Bolzano-Weierstrass, $(a_n)$ has a convergent subsequence $a_{n_k} \to L$ . One shows that the full sequence $a_n \to L$ using the Cauchy condition.

Full proof: see Cauchy Criterion Proof.

ExampleCauchy sequence converges in R

Let $a_n = \sum_{k=1}^n 1/k!$ . One can verify $(a_n)$ is Cauchy (terms eventually differ by $< \epsilon$ ). By the Cauchy criterion, $(a_n)$ converges in $\mathbb{R}$ . The limit is $e - 1 \approx 1.71828$ .

ExampleCauchy in Q need not converge in Q

Let $a_n = (1 + 1/n)^n$ (all terms rational). Then $(a_n)$ is Cauchy as a sequence in $\mathbb{Q}$ (with the usual metric). However, the limit $e \approx 2.71828\ldots$ is irrational, so $(a_n)$ does not converge in $\mathbb{Q}$ . This shows $\mathbb{Q}$ is not complete.

Properties of Cauchy sequences

Theorem2.2Cauchy sequences are bounded

Every Cauchy sequence is bounded.

Proof

$M = \max(|a_1|, |a_2|, \ldots, |a_{N-1}|, |a_N| + 1).$

Then $|a_n| \leq M$ for all $n$ .

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RemarkBounded does not imply Cauchy

A bounded sequence need not be Cauchy. For example, $a_n = (-1)^n$ is bounded but not Cauchy (terms oscillate, so $|a_{n+1} - a_n| = 2$ for all $n$ ).

Theorem2.3Algebra of Cauchy sequences

If $(a_n)$ and $(b_n)$ are Cauchy, then so are $(a_n + b_n)$ , $(a_n - b_n)$ , and $(a_n b_n)$ (assuming boundedness for the product).

ExampleSum of Cauchy sequences

If $(a_n)$ and $(b_n)$ are Cauchy, then for large $m, n$ ,

$|(a_m + b_m) - (a_n + b_n)| \leq |a_m - a_n| + |b_m - b_n| < \epsilon/2 + \epsilon/2 = \epsilon.$

Thus $(a_n + b_n)$ is Cauchy.

Completeness of R

The Cauchy criterion is equivalent to the completeness axiom for $\mathbb{R}$ .

Theorem2.4Equivalence of completeness notions

The following are equivalent for $\mathbb{R}$ :

Every nonempty bounded set has a supremum (least upper bound property).
Every Cauchy sequence converges (Cauchy completeness).
Every bounded monotone sequence converges.

RemarkProof strategy

(1) $\Rightarrow$ (3) is the Monotone Convergence Theorem. (3) $\Rightarrow$ (2) uses Bolzano-Weierstrass. (2) $\Rightarrow$ (1) constructs suprema as limits of Cauchy sequences. See Monotone Convergence Theorem and Cauchy Criterion.

ExampleR is the completion of Q

$\mathbb{R}$ can be defined as the set of equivalence classes of Cauchy sequences in $\mathbb{Q}$ (modulo sequences differing by a null sequence). This is the Cauchy completion of $\mathbb{Q}$ . Every Cauchy sequence in $\mathbb{Q}$ corresponds to a unique real number (its limit in $\mathbb{R}$ ).

Summary

Cauchy sequences provide an intrinsic characterization of convergence:

A sequence is Cauchy if terms become arbitrarily close to each other.
In $\mathbb{R}$ , Cauchy sequences are exactly the convergent sequences (Cauchy completeness).
In $\mathbb{Q}$ , Cauchy sequences need not converge — this motivates the construction of $\mathbb{R}$ .
Cauchy completeness is equivalent to the least upper bound property.

See Cauchy Criterion for the proof of the Cauchy criterion, and Metric Spaces for generalizations beyond $\mathbb{R}$ .