Sequences of Real Numbers — Convergence and Cauchy

A sequence $(a_{n})$ of real numbers converges to a limit $L$ if its terms become and stay arbitrarily close to $L$ ; it is Cauchy if its terms become arbitrarily close to each other. Key result: in $R$ these two notions are equivalent (completeness), so convergence can be verified without knowing the limit. Applies throughout analysis — series, continuity, integration, numerical methods. Formalized by Cauchy (1821) and Weierstrass (1870s).

A sequence of real numbers is a function $a : N \to R$ , written $(a_{n})_{n \geq 1}$ . Convergence $a_{n} \to L$ means that for every tolerance $ε > 0$ there is an index past which all terms lie within $ε$ of $L$ . The Cauchy criterion replaces the unknown $L$ with an internal condition: terms eventually cluster within $ε$ of one another. This matters because it lets us prove convergence without identifying the limit — essential for series, decimal expansions, and constructing $R$ itself. The surprising fact: Cauchy = convergent is false in $Q$ (e.g. truncations of $2$ ), and it is exactly this failure that the real numbers are built to repair.

The Core Insight

Convergence is a statement about the relationship between terms and an external limit point

L

. Cauchy is a statement entirely internal to the sequence. In a complete ordered field — which is precisely what

R

is — the internal condition forces the existence of the external limit. Completeness is the hidden engine of all of real analysis.

Formal Statement

Sequence

A sequence of real numbers is a function $a : N \to R$ , $n \mapsto a_{n}$ . The value $a_{n}$ is the $n$ -th term. We denote the sequence by $(a_{n})_{n \geq 1}$ or simply $(a_{n})$ . A sequence is bounded if there is $M > 0$ with $∣ a_{n} ∣ \leq M$ for all $n$ .

Convergence

A sequence $(a_{n})$ converges to $L \in R$ if

\forall ε > 0 \exists N \in N \forall n \geq N : ∣ a_{n} - L ∣ < ε .

⟳this is the formal reading of "terms get and stay within any prescribed tolerance

ε

L

N

is allowed to depend on

ε

We write $lim_{n \to \infty} a_{n} = L$ or $a_{n} \to L$ . Symbols: $ε$ = tolerance (any positive real), $N$ = threshold index, $L$ = limit, $∣ a_{n} - L ∣$ = distance from term to limit. A sequence is divergent if no such $L$ exists.

Cauchy Sequence

A sequence $(a_{n})$ is a Cauchy sequence if

\forall ε > 0 \exists N \in N \forall m, n \geq N : ∣ a_{n} - a_{m} ∣ < ε .

⟳replace the external

L

in [1] with another term

a_{m}

; the condition becomes purely internal to the sequence.

Here $∣ a_{n} - a_{m} ∣$ measures the mutual closeness of two late terms. Note the two quantified indices $m, n$ .

Cauchy Completeness of $\mathbb{R}$

A sequence of real numbers converges if and only if it is Cauchy:

(a_{n}) converges in R ⟺ (a_{n}) is Cauchy .

⟳forward direction is the triangle inequality; backward direction uses boundedness + Bolzano–Weierstrass + the completeness axiom.

This is equivalent to the Least Upper Bound Property of $R$ and characterizes $R$ as a complete ordered field.

Uniqueness of Limits

If $a_{n} \to L$ and $a_{n} \to L^{'}$ then $L = L^{'}$ .

∣ L - L^{'} ∣ \leq ∣ L - a_{n} ∣ + ∣ a_{n} - L^{'} ∣ < ε ⟹ L = L^{'} .

⟳choose

N

large enough for both convergences with tolerance

ε /2

; since

∣ L - L^{'} ∣ < ε

for every

ε

, it is

0

Why It Holds

Convergent ⟹ Cauchy

Assume $a_{n} \to L$ . Fix $ε > 0$ .

Step 1. Apply [1] with tolerance

ε /2

: there is

N

such that

∣ a_{n} - L ∣ < ε /2

for all

n \geq N

. Step 2. Take any

m, n \geq N

. By the triangle inequality,

∣ a_{n} - a_{m} ∣ \leq ∣ a_{n} - L ∣ + ∣ L - a_{m} ∣ < \frac{ε}{2} + \frac{ε}{2} = ε .

⟳insert

\pm L

inside the absolute value and split.

Step 3. Since

ε > 0

was arbitrary, [2] holds. Thus

(a_{n})

is Cauchy.

■

Cauchy ⟹ Convergent (in $\mathbb{R}$)

Assume $(a_{n})$ is Cauchy.

Step 1 — Cauchy sequences are bounded. Apply [2] with

ε = 1

: there is

N

with

∣ a_{n} - a_{N} ∣ < 1

for

n \geq N

, hence

∣ a_{n} ∣ < ∣ a_{N} ∣ + 1

. Then

∣ a_{n} ∣ \leq max {∣ a_{1} ∣, \dots, ∣ a_{N - 1} ∣, ∣ a_{N} ∣ + 1} =: M for all n .

⟳only finitely many early terms are unconstrained; take the max with the tail bound.

Step 2 — A convergent subsequence exists. By the Bolzano–Weierstrass Theorem, the bounded sequence

(a_{n})

has a subsequence

(a_{n_{k}})

converging to some

L \in R

(this step uses completeness of $R$ ). Step 3 — The whole sequence converges to $L$ . Fix

ε > 0

. By [2] get

N_{1}

with

∣ a_{n} - a_{m} ∣ < ε /2

for

m, n \geq N_{1}

. By subsequence convergence get

k

with

n_{k} \geq N_{1}

and

∣ a_{n_{k}} - L ∣ < ε /2

. Then for all

n \geq N_{1}

∣ a_{n} - L ∣ \leq ∣ a_{n} - a_{n_{k}} ∣ + ∣ a_{n_{k}} - L ∣ < \frac{ε}{2} + \frac{ε}{2} = ε .

⟳the term

a_{n_{k}}

acts as a bridge — close to all late terms by Cauchy, close to

L

by the subsequence.

Hence $a_{n} \to L$ . $■$

Completeness is essential

Step 2 is where $R$ is irreplaceable. In $Q$ , Bolzano–Weierstrass fails and there exist Cauchy sequences of rationals with no rational limit. The theorem [3] is a property of $R$ , not a logical tautology about sequences.

Augustin-Louis Cauchy | 1789–1857

Cauchy introduced the rigorous $ε$ -style treatment of limits in his Cours d'analyse (1821) and Résumé des leçons (1823). He formulated the criterion now bearing his name to test convergence of series without summing them, though he (and his era) lacked an axiomatic construction of $R$ and so implicitly assumed completeness. The gap was closed by Weierstrass, Dedekind, and Cantor in the 1870s, who constructed the reals precisely so that Cauchy's criterion would be a theorem rather than an assumption.

Building Intuition

Picture the terms as dots landing on a number line as $n$ grows.

Convergence says: draw any tiny band of half-width $ε$ around the target $L$ . Eventually all dots fall inside that band and never leave.
Cauchy says: eventually all the dots are within $ε$ of each other — they huddle into an arbitrarily small cluster, regardless of whether you've marked a target.

The miracle of $R$ : a cluster that keeps shrinking must be shrinking toward a point that genuinely exists. There are no "holes" for the cluster to vanish into. In $Q$ there are holes (the irrationals), so a shrinking rational cluster can converge to a missing point like $2$ .

Convergence band vs. Cauchy clustering

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Worked Examples

A convergent sequence — $a_n = \tfrac{3n+1}{n}$

Given:

a_{n} = \frac{3 n + 1}{n} = 3 + \frac{1}{n}

. Claim

L = 3

. Step 1. Compute the distance:

∣ a_{n} - 3∣ = 3 + \frac{1}{n} - 3 = \frac{1}{n}

. Step 2. Demand

\frac{1}{n} < ε

, i.e.

n > \frac{1}{ε}

. Choose

N = ⌊ \frac{1}{ε} ⌋ + 1.

⟳solve

1/ n < ε

for

n

, then round up to the next integer to guarantee strictness.

Step 3. For all

n \geq N

we have

n > 1/ ε

, hence

∣ a_{n} - 3∣ = 1/ n < ε

, satisfying [1]. Numerical check: for

ε = 0.01

N = ⌊ 100 ⌋ + 1 = 101

. Then

a_{101} = 3 + 1/101 \approx 3.0099

, distance

\approx 0.0099 < 0.01

. ✓

n \to \infty lim \frac{3 n + 1}{n} = 3

Verifying Cauchy directly — $a_n = \sum_{k=1}^n \tfrac{1}{2^k}$

Given: partial sums

a_{n} = \frac{1}{2} + \frac{1}{4} + \dots + \frac{1}{2 ^{n}} = 1 - \frac{1}{2 ^{n}}

. Step 1. For

n > m

compute the gap:

∣ a_{n} - a_{m} ∣ = 1 - \frac{1}{2 ^{n}} - (1 - \frac{1}{2 ^{m}}) = \frac{1}{2 ^{m}} - \frac{1}{2 ^{n}} < \frac{1}{2 ^{m}} .

⟳the geometric tail telescopes; bound it by its leading term.

Step 2. Demand

\frac{1}{2 ^{m}} < ε

, i.e.

2^{m} > \frac{1}{ε}

, i.e.

m > lo g_{2} (1/ ε)

. Choose

N = ⌊ lo g_{2} (1/ ε)⌋ + 1

. Step 3. For all

m, n \geq N

(say

n > m

∣ a_{n} - a_{m} ∣ < 1/ 2^{m} \leq 1/ 2^{N} < ε

. Condition [2] holds, so

(a_{n})

is Cauchy, hence convergent by [3] without computing the limit. Numerical check:

ε = 0.001 \Rightarrow N = ⌊ lo g_{2} 1000 ⌋ + 1 = ⌊ 9.97 ⌋ + 1 = 10

. Indeed

1/ 2^{10} = 0.000977 < 0.001

. ✓ (The limit happens to be

1

(a_{n}) is Cauchy, hence convergent; lim a_{n} = 1

Common Misconceptions

Consecutive terms close ≠ Cauchy

A frequent error: believing $∣ a_{n + 1} - a_{n} ∣ \to 0$ implies $(a_{n})$ is Cauchy. It does not. The Cauchy condition [2] requires all pairs $m, n \geq N$ to be close, not just adjacent ones. The harmonic partial sums $H_{n} = \sum_{k = 1}^{n} 1/ k$ satisfy $∣ H_{n + 1} - H_{n} ∣ = 1/ (n + 1) \to 0$ yet $(H_{n})$ diverges to $+ \infty$ .

Cauchy in $\mathbb{Q}$ that fails to converge in $\mathbb{Q}$

Define rational truncations of $2$ : $a_{1} = 1.4, a_{2} = 1.41, a_{3} = 1.414, \dots$ , each the decimal expansion of $2$ to $n$ places.

∣ a_{n} - a_{m} ∣ \leq 1 0^{- m i n (m, n)} m, n \to \infty 0,

⟳two truncations agreeing to

min (m, n)

places differ only beyond that decimal place.

so $(a_{n})$ is Cauchy. Its only possible limit is $2 \in / Q$ . Thus inside $Q$ it is Cauchy but divergent — the field $Q$ is not complete. This is exactly the defect that the construction of $R$ via Cauchy sequences (Cantor) or cuts (Dedekind) repairs.

Bounded but not convergent

a_{n} = (- 1)^{n}

is bounded (

∣ a_{n} ∣ = 1

) but not convergent:

∣ a_{n + 1} - a_{n} ∣ = 2

for all

n

, violating [2]. Boundedness alone is insufficient; one needs Cauchy (or monotone + bounded).

Connections

Algebra of limits. If $a_{n} \to A$ , $b_{n} \to B$ then $a_{n} \pm b_{n} \to A \pm B$ , $a_{n} b_{n} \to A B$ , and $a_{n} / b_{n} \to A / B$ (if $B \neq = 0$ ). These follow from [1] plus the triangle inequality — see Limit Laws for Sequences.
Monotone Convergence. A monotone bounded sequence converges; this is logically equivalent to [3] and to the Least Upper Bound Property. See Monotone Convergence Theorem.
Series. A series $\sum a_{k}$ converges iff its partial sums $S_{n} = \sum_{k \leq n} a_{k}$ form a Cauchy sequence — the Cauchy Criterion for Series is just [2] applied to $(S_{n})$ .
Subsequences. Every subsequence of a convergent sequence converges to the same limit; conversely Bolzano–Weierstrass Theorem extracts convergent subsequences from bounded ones.
Topology. Cauchy and convergence are unified by Complete Metric Spaces, where [3] becomes the definition of completeness.
limsup/liminf. Convergence is equivalent to $lim inf a_{n} = lim sup a_{n} \in R$ ; see Limit Superior and Limit Inferior.

Real-World Applications

Numerical root-finding. Newton's method generates an iterate sequence $x_{n + 1} = x_{n} - f (x_{n}) / f^{'} (x_{n})$ ; proving it is Cauchy (via a contraction estimate) guarantees convergence to a root — see Banach Fixed Point Theorem.
IEEE-754 floating-point summation. Algorithms like Kahan summation rely on knowing that partial sums of a convergent series form a tightening Cauchy cluster, bounding accumulated round-off.
Construction of $R$ (Cantor, 1872). The real numbers are defined as equivalence classes of Cauchy sequences of rationals — completeness is built in by fiat.
Fourier series convergence. Partial Fourier sums are tested for $L^{2}$ convergence via Cauchy criteria in Hilbert Spaces, underpinning signal processing and JPEG/MP3 compression.
Markov chain Monte Carlo (MCMC). Convergence diagnostics check whether sample averages form an (approximately) Cauchy/convergent sequence before trusting estimates.
Iterative PDE solvers. Finite-difference and Gauss–Seidel iterations are stopped when successive approximations satisfy a Cauchy-type tolerance $∥ x_{n} - x_{m} ∥ < ε$ .
Continued fractions. Convergents of $2$ , $π$ , or the golden ratio form Cauchy sequences whose limits are the irrational targets, used in clock-rate approximation and gear-ratio design.

Historical Development

c. 450 BCE — Zeno of Elea: paradoxes of motion implicitly probe infinite sums and convergence.
c. 1660s — Newton & Leibniz: calculus uses limiting processes informally ("ultimate ratios," infinitesimals) without rigor.
1734 — Berkeley's The Analyst: attacks the logical foundations ("ghosts of departed quantities"), spurring demand for rigor.
1821 — Cauchy, Cours d'analyse: introduces a rigorous treatment of limits and convergence; states the Cauchy criterion for series.
1817 / rediscovered 1860s — Bolzano: proves what became the Bolzano–Weierstrass theorem and an $ε$ -style continuity, but his work was largely unread for decades.
1854–1865 — Weierstrass (lectures): crystallizes the modern $ε$ – $N$ / $ε$ – $δ$ definitions, removing reliance on intuition and infinitesimals.
1872 — Cantor & Dedekind: independently construct $R$ — Cantor via Cauchy sequences of rationals, Dedekind via cuts — turning completeness into a theorem.
1872 — Heine: systematizes uniform continuity and works closely with Cantor's framework, popularizing the sequence-based approach.
1906 — Fréchet: abstracts these ideas into metric spaces, generalizing Cauchy/convergence beyond $R$ into Complete Metric Spaces.

Subtleties & Limits

The equivalence [3] is special to complete spaces; deciding completeness or constructing completions raises deep issues.

Effective/constructive content. A classically Cauchy sequence may have no computable rate of convergence (a "modulus"). Constructive mathematics (Bishop) therefore defines Cauchy with an explicit modulus $N (ε)$ , and not every classical theorem survives.
Order of completion. $R$ is the metric completion of $Q$ under $∣ \cdot ∣$ . Choosing a different metric — the $p$ -adic absolute value — completes $Q$ to the totally different field $Q_{p}$ . "Cauchy" is metric-dependent.
Rate vs. existence. Knowing a sequence converges (via Cauchy) gives no a priori bound on how fast; quantitative analysis and proof mining (Kohlenbach) extract explicit rates from classical existence proofs — an active research program.
Beyond metrics. In general topological spaces "Cauchy" is not even well-defined without a uniformity; nets and filters replace sequences, and convergence/completeness become genuinely subtler.

Sequences of Real Numbers — Convergence and Cauchy

The Core Insight

Formal Statement

Sequence

Convergence

Cauchy Sequence

Cauchy Completeness of $\mathbb{R}$

Uniqueness of Limits

Why It Holds

Convergent ⟹ Cauchy

Cauchy ⟹ Convergent (in $\mathbb{R}$)

Completeness is essential

Augustin-Louis Cauchy | 1789–1857

Building Intuition

Convergence band vs. Cauchy clustering

Worked Examples

A convergent sequence — $a_n = \tfrac{3n+1}{n}$

Verifying Cauchy directly — $a_n = \sum_{k=1}^n \tfrac{1}{2^k}$

Common Misconceptions

Consecutive terms close ≠ Cauchy

Cauchy in $\mathbb{Q}$ that fails to converge in $\mathbb{Q}$

Bounded but not convergent

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also

Before this

Where this leads