Sequences: Convergence, Cauchy Sequences, Subsequences

A sequence $(a_n)$ converges to a limit $L$ if its terms get arbitrarily close to $L$ for all large $n$; it is Cauchy if its terms get arbitrarily close to each other. In $\mathbb{R}$ (and any complete space) these two notions coincide. Subsequences are selections of terms in order, and the Bolzano–Weierstrass theorem guarantees every bounded real sequence has a convergent subsequence. These ideas, formalized by Cauchy and Weierstrass in the 19th century, are the bedrock of analysis.

A sequence is a function from the natural numbers into a set, written $(a_n{)}_{n=1}^{\infty }$. Convergence captures the precise sense in which a sequence "approaches" a number, replacing vague appeals to motion with the static $\epsilon$–$N$ definition. The Cauchy criterion lets us prove convergence without knowing the limit in advance — a profound conceptual leap, since it characterizes convergence by an internal property of the sequence alone. A surprising fact: in $\mathbb{Q}$, Cauchy sequences need not converge (e.g. rational approximations to $\sqrt{2}$), and it is precisely the demand that all Cauchy sequences converge that defines the real numbers as the completion of the rationals.

The Core Insight

Convergence is an external condition (closeness to a fixed target $L$), while the Cauchy condition is an internal one (mutual closeness of terms). The theorem that the two are equivalent — completeness — is what makes $\mathbb{R}$ a usable arena for analysis. Subsequences let us extract order out of chaos: even when a sequence wanders, boundedness forces some thread of it to settle down.

Formal Statement

Sequence

A sequence in a set $X$ is a function $a:\mathbb{N}\to X$, denoted $(a_n{)}_{n\ge 1}$ where $a_n=a(n)$. We mostly take $X=\mathbb{R}$ or $X=\mathbb{C}$. The range $\{a_n:n\in \mathbb{N}\}$ is a set (may be finite), while the sequence itself is an ordered, possibly-repeating list. A sequence is bounded if there exists $M>0$ with $|a_n|\le M$ for all $n$.

Convergence ($\varepsilon$–$N$)

A sequence $(a_n)$ in $\mathbb{R}$ converges to $L\in \mathbb{R}$, written $a_n\to L$ or ${\lim }_{n\to \infty }{a}_n=L$, if:

Here $\epsilon$ is an arbitrary positive tolerance, $N$ the threshold index (depends on $\epsilon$), and $|a_n-L|$ the distance from term to limit. A sequence that does not converge diverges.

Cauchy Sequence

A sequence $(a_n)$ is Cauchy if:

Subsequence

Given $(a_n)$ and a strictly increasing function $k\mapsto n_k$ (so $n_1<n_2<n_3<\cdots$, all in $\mathbb{N}$), the sequence

is a subsequence of $(a_n)$. A point $L$ is a subsequential limit (or cluster point) if some subsequence converges to $L$.

Fundamental Theorems on Sequences

Let $(a_n)$ be a real sequence.

1. (Uniqueness) If $a_n\to L$ and $a_n\to L^{\prime }$, then $L=L^{\prime }$.
2. (Convergent $\Rightarrow$ bounded) If $(a_n)$ converges, it is bounded.
3. (Convergent $\Rightarrow$ Cauchy) Every convergent sequence is Cauchy.
4. (Cauchy $\Rightarrow$ bounded) Every Cauchy sequence is bounded.
5. (Completeness of $\mathbb{R}$) Every Cauchy sequence in $\mathbb{R}$ converges.
6. (Monotone Convergence) A monotone bounded sequence converges (to its sup if increasing, inf if decreasing).
7. (Bolzano–Weierstrass) Every bounded real sequence has a convergent subsequence.
8. (Subsequence inheritance) If $a_n\to L$, then every subsequence $a_{n_k}\to L$.
9. (Cluster characterization) $a_n\to L$ iff $(a_n)$ is bounded and $L$ is its only subsequential limit.

Why It Holds

Convergent $\Rightarrow$ Cauchy (Theorem 3)

1. Suppose $a_n\to L$. Fix $\epsilon >0$.
2. Apply [1] with tolerance $\epsilon /2$: there is $N$ such that $n\ge N\Rightarrow |a_n-L|<\epsilon /2$.
3. Take any $m,n\ge N$. By the triangle inequality:

1. Thus [2] holds, so $(a_n)$ is Cauchy. $■$

Bolzano–Weierstrass (Theorem 7) via bisection

1. Let $(a_n)$ satisfy $|a_n|\le M$, so all terms lie in $I_0=[-M,M]$.
2. Bisect $I_0$ into two halves; at least one half contains $a_n$ for infinitely many $n$. Call it $I_1$; pick $n_1$ with $a_{n_1}\in I_1$.
3. Inductively bisect $I_k$, choosing the half $I_{k+1}$ containing infinitely many terms, and pick $n_{k+1}>n_k$ with $a_{n_{k+1}}\in I_{k+1}$. This is possible because $I_{k+1}$ holds infinitely many terms.
4. The interval lengths satisfy

1. The nested closed intervals $I_0\supseteq I_1\supseteq \cdots$ with lengths $\to 0$ have, by the Nested Interval Property, a unique common point $L={\bigcap }_k{I}_k$.
2. Since $a_{n_k},L\in I_k$, we get $|a_{n_k}-L|\le |I_k|\to 0$, so $a_{n_k}\to L$. $■$

Cauchy $\Rightarrow$ Convergent in $\mathbb{R}$ (Theorem 5)

1. Let $(a_n)$ be Cauchy. By Theorem 4 it is bounded.
2. By Bolzano–Weierstrass it has a subsequence $a_{n_k}\to L$ for some $L$.
3. Fix $\epsilon >0$. Choose $N_1$ so that $m,n\ge N_1\Rightarrow |a_m-a_n|<\epsilon /2$ (Cauchy). Choose $k$ large so that $n_k\ge N_1$ and $|a_{n_k}-L|<\epsilon /2$ (subsequence limit).
4. For any $n\ge N_1$:

1. Hence $a_n\to L$. The boundedness in step 1 plus B–W requires the completeness of $\mathbb{R}$ (Nested Interval Property), which fails in $\mathbb{Q}$. $■$

Cauchy $\Rightarrow$ bounded (Theorem 4)

1. Apply [2] with $\epsilon =1$: there is $N$ with $|a_m-a_n|<1$ for $m,n\ge N$.
2. In particular $|a_n-a_N|<1$, so $|a_n|<|a_N|+1$ for all $n\ge N$.
3. Let $M=\max \{|a_1|,\dots ,|a_{N-1}|,|a_N|+1\}$. Then $|a_n|\le M$ for all $n$. $■$

Augustin-Louis Cauchy | 1789–1857

Cauchy gave analysis its modern rigor in his Cours d'analyse (1821), introducing the convergence criterion now bearing his name and insisting on $\epsilon$-style bounds where his predecessors relied on intuition about "infinitely small" quantities. Though his own proof that Cauchy sequences converge implicitly assumed completeness of $\mathbb{R}$ (a property not yet axiomatized), his formulation isolated exactly the right internal condition. Weierstrass and later Cantor and Dedekind supplied the missing foundation by constructing $\mathbb{R}$ so that Cauchy sequences would necessarily converge.

Building Intuition

Think of convergence as a homing missile: $L$ is the target, and for any shrinking circle of radius $\epsilon$ around it, the sequence eventually enters and never leaves. The Cauchy condition removes the target: instead, you watch the terms huddle together — past some point they all crowd into a region of diameter $\epsilon$. In $\mathbb{R}$, huddling guarantees a home: there's always a real number for the huddle to converge to. In $\mathbb{Q}$, the huddle can form around a "hole" (an irrational), and no rational target exists — the sequence is Cauchy but stranded. A subsequence is choosing which terms to watch, in order, skipping freely; Bolzano–Weierstrass says a bounded sequence may bounce around, but you can always pick out a calm thread that settles.

Convergence funnel: terms enter the $\varepsilon$-band past threshold $N$

Worked Examples

Proving $a_n = \tfrac{2n+1}{n+3}\to 2$ by $\varepsilon$–$N$

A Cauchy sequence in $\mathbb{Q}$ with no rational limit

Common Misconceptions

Consecutive differences vanish, yet the sequence diverges

Let $a_n={\sum }_{k=1}^n\frac{1}{k}$ (harmonic partial sums). Then

Since $|a_{2n}-a_n|\nrightarrow 0$, $(a_n)$ is not Cauchy, hence diverges (in fact $a_n\to \infty$). The vanishing of consecutive gaps is far too weak.

A bounded divergent sequence — with two subsequential limits

Connections

Sequence convergence is the engine beneath nearly all of analysis. Series and Convergence Tests defines a series as the sequence of partial sums, so every convergence test is secretly a sequence statement. Limits of Functions can be characterized sequentially (Heine's criterion: $f(x)\to L$ as $x\to a$ iff $f(x_n)\to L$ for every sequence $x_n\to a$). Continuity then becomes "sequence-preserving." The Cauchy criterion generalizes to Metric Spaces, where completeness distinguishes well-behaved spaces, and to Banach Spaces (complete normed vector spaces) underlying functional analysis. Bolzano–Weierstrass is the prototype of sequential compactness, linking to Compactness and the Heine–Borel theorem.

Real-World Applications

1. Numerical fixed-point iteration (Newton's method, Banach fixed-point theorem): convergence and Cauchy estimates certify when an iterative solver $x_{n+1}=g(x_n)$ reaches a root, with error bounds $|x_n-x^{\ast }|\le \frac{c^n}{1-c}|x_1-x_0|$.
2. Constructing the real numbers (Cantor's completion): $\mathbb{R}$ is defined as equivalence classes of Cauchy sequences of rationals — the entire number system used in physics and engineering rests on this.
3. Floating-point convergence criteria in software (e.g. SciPy, MATLAB solvers) use $|x_n-x_{n-1}|<\text{tol}$ as a practical Cauchy stopping rule — with the caveat from the warning above about its limits.
4. Signal processing / Fourier analysis: convergence of partial sums (a sequence of functions) determines reconstruction fidelity; Cauchy completeness of $L^2$ underwrites the orthonormal expansion machinery.
5. Markov chain Monte Carlo (MCMC) convergence diagnostics (Gelman–Rubin) monitor whether sampled subsequences from multiple chains settle to a common stationary distribution.
6. Control theory / Lyapunov stability: trajectories of discrete-time systems are sequences whose convergence to an equilibrium certifies asymptotic stability.
7. Machine learning optimization: gradient descent produces a parameter sequence; convergence theorems guarantee (under convexity/step-size conditions) that the loss sequence is Cauchy and converges to a minimizer.

Historical Development

• c. 450 BCE — Zeno's paradoxes implicitly probe infinite sequences and sums, foreshadowing limit puzzles.
• c. 1665 — Newton and Leibniz use convergent series and "ultimate ratios" in calculus, without rigorous limits.
• 1734 — Berkeley's The Analyst attacks the logical vagueness of infinitesimals ("ghosts of departed quantities"), exposing the need for rigor.
• 1817 — Bolzano proves (in obscurity) an intermediate-value-style result and the essence of what becomes the Bolzano–Weierstrass theorem.
• 1821 — Cauchy's Cours d'analyse formalizes convergence and the Cauchy criterion, anchoring analysis in inequalities.
• 1872 — Weierstrass, Cantor, Dedekind, and Heine independently rigorize the real numbers; Cantor builds $\mathbb{R}$ from Cauchy sequences; Dedekind from cuts.
• 1872 — Weierstrass popularizes the $\epsilon$–$\delta$ / $\epsilon$–$N$ formalism in lectures, the form taught today.
• 1906 — Fréchet introduces abstract metric spaces, generalizing Cauchy sequences and completeness beyond $\mathbb{R}$.
• 1922 — Banach's thesis establishes complete normed spaces and the contraction mapping principle, embedding sequence convergence at the heart of functional analysis.

Subtleties & Limits

See Also