Sequences of Real Numbers — Convergence and Cauchy

⚡ Key Takeaways

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 $\mathbb{R}$ 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).

💡 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 $\mathbb{R}$ is — the internal condition forces the existence of the external limit. Completeness is the hidden engine of all of real analysis.

DefinitionSequence

A sequence of real numbers is a function $a:\mathbb{N}\to\mathbb{R}$ , $n\mapsto a_n$ . The value $a_n$ is the $n$ -th term. We denote the sequence by $(a_n)_{n\ge1}$ or simply $(a_n)$ . A sequence is bounded if there is $M>0$ with $|a_n|\le M$ for all $n$ .

DefinitionConvergence

A sequence $(a_n)$ converges to $L\in\mathbb{R}$ if

1

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

Derivation hint: this is the formal reading of "terms get and stay within any prescribed tolerance $\varepsilon$ of $L$ "; $N$ is allowed to depend on $\varepsilon$ . We write $\lim_{n\to\infty}a_n = L$ or $a_n\to L$ . Symbols: $\varepsilon$ = 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.

DefinitionCauchy Sequence

A sequence $(a_n)$ is a Cauchy sequence if

2

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

Derivation hint: 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 $m,n$ .

TheoremCauchy Completeness of

\mathbb{R}

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

3

\quad (a_n)\ \text{converges in }\mathbb{R}\ \iff\ (a_n)\ \text{is Cauchy}.

Derivation hint: 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 $\mathbb{R}$ and characterizes $\mathbb{R}$ as a complete ordered field.

TheoremUniqueness of Limits

If $a_n\to L$ and $a_n\to L'$ then $L=L'$ .

4

\quad |L-L'|\le |L-a_n|+|a_n-L'|<\varepsilon \quad\Longrightarrow\quad L=L'.

Derivation hint: choose $N$ large enough for both convergences with tolerance $\varepsilon/2$ ; since $|L-L'|<\varepsilon$ for every $\varepsilon$ , it is $0$ .

ProofConvergent ⟹ Cauchy

Assume $a_n\to L$ . Fix $\varepsilon>0$ .

Step 1. Apply [1] with tolerance $\varepsilon/2$ : there is $N$ such that $|a_n-L|<\varepsilon/2$ for all $n\ge N$ .

Step 2. Take any $m,n\ge N$ . By the triangle inequality,

5

\quad |a_n-a_m| \le |a_n-L| + |L-a_m| < \tfrac{\varepsilon}{2} + \tfrac{\varepsilon}{2} = \varepsilon.

Derivation hint: insert $\pm L$ inside the absolute value and split.

Step 3. Since $\varepsilon>0$ was arbitrary, [2] holds. Thus $(a_n)$ is Cauchy. $\blacksquare$

∎

ProofCauchy ⟹ Convergent (in

\mathbb{R}

)

Assume $(a_n)$ is Cauchy.

Step 1 — Cauchy sequences are bounded. Apply [2] with $\varepsilon=1$ : there is $N$ with $|a_n-a_N|<1$ for $n\ge N$ , hence $|a_n|<|a_N|+1$ . Then

6

\quad |a_n|\le \max\{\,|a_1|,\dots,|a_{N-1}|,\ |a_N|+1\,\}=:M \quad\text{for all }n.

Derivation hint: 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 $L\in\mathbb{R}$ (this step uses completeness of $\mathbb{R}$ ).

Step 3 — The whole sequence converges to $L$ . Fix $\varepsilon>0$ . By [2] get $N_1$ with $|a_n-a_m|<\varepsilon/2$ for $m,n\ge N_1$ . By subsequence convergence get $k$ with $n_k\ge N_1$ and $|a_{n_k}-L|<\varepsilon/2$ . Then for all $n\ge N_1$ ,

7

\quad |a_n-L|\le |a_n-a_{n_k}| + |a_{n_k}-L| < \tfrac{\varepsilon}{2}+\tfrac{\varepsilon}{2}=\varepsilon.

Derivation hint: the term $a_{n_k}$ acts as a bridge — close to all late terms by Cauchy, close to $L$ by the subsequence. Hence $a_n\to L$ . $\blacksquare$

∎

⚠️ Completeness is essential

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

mathematician · Augustin-Louis Cauchy | 1789–1857

Cauchy introduced the rigorous $\varepsilon$ -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 $\mathbb{R}$ 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.

🧠 Intuition

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

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

The miracle of $\mathbb{R}$ : 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 $\mathbb{Q}$ there are holes (the irrationals), so a shrinking rational cluster can converge to a missing point like $\sqrt2$ .

Figure

Convergence band vs. Cauchy clustering

```svg

```

ExampleA convergent sequence —

a_n = \tfrac{3n+1}{n}

Given: $a_n = \dfrac{3n+1}{n}=3+\dfrac1n$ . Claim $L=3$ .

Step 1. Compute the distance: $|a_n - 3| = \left|3+\tfrac1n-3\right| = \dfrac1n$ .

Step 2. Demand $\dfrac1n<\varepsilon$ , i.e. $n>\dfrac1\varepsilon$ . Choose

8

\quad N = \left\lfloor \tfrac1\varepsilon\right\rfloor + 1.

Derivation hint: solve $1/n<\varepsilon$ for $n$ , then round up to the next integer to guarantee strictness.

Step 3. For all $n\ge N$ we have $n>1/\varepsilon$ , hence $|a_n-3|=1/n<\varepsilon$ , satisfying [1].

Numerical check: for $\varepsilon = 0.01$ , $N = \lfloor 100\rfloor+1 = 101$ . Then $a_{101}=3+1/101\approx 3.0099$ , distance $\approx 0.0099<0.01$ . ✓

\boxed{\ \lim_{n\to\infty}\frac{3n+1}{n} = 3\ }

ExampleVerifying Cauchy directly —

a_n = \sum_{k=1}^n \tfrac{1}{2^k}

Given: partial sums $a_n = \dfrac12+\dfrac14+\cdots+\dfrac1{2^n} = 1-\dfrac1{2^n}$ .

Step 1. For $n>m$ compute the gap:

9

\quad |a_n - a_m| = \left|\,1-\tfrac1{2^n} - \left(1-\tfrac1{2^m}\right)\right| = \frac1{2^m}-\frac1{2^n} < \frac1{2^m}.

Derivation hint: the geometric tail telescopes; bound it by its leading term.

Step 2. Demand $\dfrac1{2^m}<\varepsilon$ , i.e. $2^m>\dfrac1\varepsilon$ , i.e. $m>\log_2(1/\varepsilon)$ . Choose $N=\lfloor \log_2(1/\varepsilon)\rfloor+1$ .

Step 3. For all $m,n\ge N$ (say $n>m$ ), $|a_n-a_m|<1/2^m\le 1/2^N<\varepsilon$ . Condition [2] holds, so $(a_n)$ is Cauchy, hence convergent by [3] without computing the limit.

Numerical check: $\varepsilon=0.001 \Rightarrow N = \lfloor\log_2 1000\rfloor+1 = \lfloor 9.97\rfloor+1 = 10$ . Indeed $1/2^{10}=0.000977<0.001$ . ✓ (The limit happens to be $1$ .)

\boxed{\ (a_n)\ \text{is Cauchy, hence convergent; }\ \lim a_n = 1\ }

⚠️ 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 $m,n\ge N$ to be close, not just adjacent ones. The harmonic partial sums $H_n=\sum_{k=1}^n 1/k$ satisfy $|H_{n+1}-H_n|=1/(n+1)\to0$ yet $(H_n)$ diverges to $+\infty$ .

CounterexampleCauchy in

\mathbb{Q}

that fails to converge in

\mathbb{Q}

Define rational truncations of $\sqrt2$ : $a_1=1.4,\ a_2=1.41,\ a_3=1.414,\dots$ , each the decimal expansion of $\sqrt2$ to $n$ places.

10

\quad |a_n-a_m| \le 10^{-\min(m,n)}\ \xrightarrow{\ m,n\to\infty\ }\ 0,

Derivation hint: two truncations agreeing to $\min(m,n)$ places differ only beyond that decimal place. so $(a_n)$ is Cauchy. Its only possible limit is $\sqrt2\notin\mathbb{Q}$ . Thus inside $\mathbb{Q}$ it is Cauchy but divergent — the field $\mathbb{Q}$ is not complete. This is exactly the defect that the construction of $\mathbb{R}$ via Cauchy sequences (Cantor) or cuts (Dedekind) repairs.

CounterexampleBounded 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).

connection

Algebra of limits. If $a_n\to A$ , $b_n\to B$ then $a_n\pm b_n\to A\pm B$ , $a_nb_n\to AB$ , and $a_n/b_n\to A/B$ (if $B\ne0$ ). 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 $S_n=\sum_{k\le n}a_k$ 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 $\liminf a_n=\limsup a_n\in\mathbb{R}$ ; see Limit Superior and Limit Inferior.

🔬 Where this appears

⏳ History

- 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

\varepsilon

-style continuity, but his work was largely unread for decades.

- 1854–1865 — Weierstrass (lectures): crystallizes the modern

\varepsilon

–

N

/

\varepsilon

–

\delta

definitions, removing reliance on intuition and infinitesimals.

- 1872 — Cantor & Dedekind: independently construct

\mathbb{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

\mathbb{R}

into Complete Metric Spaces.

open-problem

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 $N(\varepsilon)$ , and not every classical theorem survives.
Order of completion. $\mathbb{R}$ is the metric completion of $\mathbb{Q}$ under $|\cdot|$ . Choosing a different metric — the $p$ -adic absolute value — completes $\mathbb{Q}$ to the totally different field $\mathbb{Q}_p$ . "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

Formal Statement

Why It Holds

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also