Sequences — convergence and limits

Intuition Beginner

A sequence is an infinite list of real numbers, one entry for each counting number: $a_1,a_2,a_3,\dots$. The list never ends. The question that drives this whole chapter is whether the numbers in the list eventually settle down toward a single target value, or whether they keep wandering forever.

Take the list $1,1/2,1/3,1/4,1/5,\dots$ where the $n$-th entry is $1/n$. The entries shrink as you read down the list. Pick any tiny positive target distance, say $0.001$. Once you read past entry $1000$, every remaining entry is smaller than $0.001$. Pick a tinier distance, say $0.000001$. Once you read past entry one million, every remaining entry is smaller still. The list settles toward $0$, and we say the sequence converges to $0$.

Compare the list $1,-1,1,-1,1,-1,\dots$ where the entries flip sign forever. No matter how far you read, the entries never settle. This sequence does not converge.

The precise rule for "settles toward a target" is the heart of this unit. The rule turns the vague idea of approaching into a checkable test: name any positive tolerance, however small, and there is a point in the list past which every entry sits within that tolerance of the target. That single rule, written down carefully, is what makes the rest of calculus possible.

Visual Beginner

The picture shows a number line from $0$ to $1$. Tick marks appear at the sequence entries $1,1/2,1/3,1/4,1/5,1/10$, clustering ever more tightly around $0$. A shaded band of width $0.001$ sits around $0$, and every tick mark past entry $1000$ falls inside the band.

The band is the tolerance made visible. The picture says: name a band width, and there is an index past which every entry lives inside the band. That index is the cutoff the definition asks for.

Worked example Beginner

We check that the sequence $a_n=1/n$ converges to $0$ using the tolerance rule with a specific tolerance $\epsilon =0.001$.

The rule asks for a cutoff index $N$ such that every entry $a_n$ with index $n$ at least $N$ lies within distance $0.001$ of the target $0$. That requirement reads $1/n<0.001$.

The inequality $1/n<0.001$ rearranges to $n>1000$. So $N=1000$ works: for every index $n$ at least $1000$, the value $1/n$ is at most $1/1000=0.001$. The entry $a_{1000}=0.001$ and every later entry is smaller, so all sit within the tolerance band around $0$.

The same calculation produces a cutoff for any positive tolerance. For tolerance $0.0001$, choose $N=10000$. For tolerance $0.0000007$, choose $N=2000000$ (rounded up). Each tolerance yields its own cutoff, and a cutoff exists for every positive tolerance. That existence is exactly what the statement "the sequence converges to $0$" means.

Check your understanding Beginner

Formal definition Intermediate+

Fix the real line $\mathbb{R}$ with its absolute value and the usual order from 02.02.01. A sequence of real numbers is a function $a:\mathbb{N}\to \mathbb{R}$, written $(a_n{)}_{n\ge 1}$ or simply $(a_n)$, where $a_n=a(n)$ is the $n$-th term.

Definition (convergence). The sequence $(a_n)$ converges to $L\in \mathbb{R}$ iff for every $\epsilon >0$ there exists $N\in \mathbb{N}$ such that for every $n\ge N$, $|a_n-L|<\epsilon$. Write $a_n\to L$ or ${\lim }_{n\to \infty }{a}_n=L$.

Definition (divergence). A sequence that does not converge is divergent. Divergence covers sequences that escape to infinity ($a_n=n$), sequences that oscillate without settling ($a_n=(-1{)}^n$), and sequences with several accumulating values.

Definition (bounded sequence). A sequence $(a_n)$ is bounded iff some $M>0$ satisfies $|a_n|\le M$ for every $n\in \mathbb{N}$.

Definition (monotone sequence). $(a_n)$ is monotone increasing iff $a_n\le a_{n+1}$ for every $n$; monotone decreasing iff $a_n\ge a_{n+1}$ for every $n$; monotone if either holds.

Definition (subsequence). A subsequence of $(a_n)$ is a sequence $(a_{n_k}{)}_{k\ge 1}$ where $n_1<n_2<n_3<\cdots$ is a strictly increasing sequence of indices in $\mathbb{N}$.

Definition (Cauchy sequence). A sequence $(a_n)$ is Cauchy iff for every $\epsilon >0$ there exists $N\in \mathbb{N}$ such that $m,n\ge N$ implies $|a_m-a_n|<\epsilon$.

The Cauchy definition does not name a limit. Its equivalence with convergence over $\mathbb{R}$ is a deep theorem that depends on the completeness axiom of 02.02.01; that equivalence, together with the Bolzano-Weierstrass theorem, is proved in full in 02.03.02. This unit builds the rest of the convergence toolkit — the limit laws, the squeeze theorem, the monotone convergence theorem — directly on the ε–$N$ definition, and states the Cauchy and Bolzano-Weierstrass results as forward references.

The presentation follows Rudin ^{[Rudin Ch. 3]} and Abbott ^{[Abbott Ch. 2–3]}; both organise the chapter so that the ε–$N$ definition and the algebraic limit theorem precede subsequences and the Cauchy criterion.

Counterexamples to common slips

• Consecutive entries getting closer to $L$ does not imply convergence to $L$. The sequence $a_n=1+1/n$ has entries whose distance to $0$ decreases at every step, yet $a_n\to 1$. Convergence requires the distance to drop below every positive tolerance, not merely to decrease.
• A bounded sequence need not converge. The sequence $a_n=(-1{)}^n$ is bounded by $1$ but oscillates between two values. Boundedness plus monotonicity is what forces convergence (the monotone convergence theorem below); boundedness alone gives only a convergent subsequence (Bolzano-Weierstrass, 02.03.02).
• A sequence can have several subsequential limits without converging. The sequence $a_n=(-1{)}^n$ has a subsequence converging to $1$ and another converging to $-1$. The full sequence converges iff every subsequence converges to the same limit.
• Strict inequalities are not preserved by limits. If $a_n<b_n$ for every $n$, the limits may agree: take $a_n=0$ and $b_n=1/n$. The limit preserves only non-strict inequality ($a_n\le b_n\Rightarrow \lim a_n\le \lim b_n$), proved below.

Key theorem with proof Intermediate+

Proposition (uniqueness of limits). If $a_n\to L$ and $a_n\to L^{\prime }$ in $\mathbb{R}$, then $L=L^{\prime }$.

Proof. Suppose $L\ne L^{\prime }$. Set $\epsilon =|L-L^{\prime }|/2>0$. Choose $N$ so that $n\ge N$ gives both $|a_n-L|<\epsilon$ and $|a_n-L^{\prime }|<\epsilon$. The triangle inequality of 00.01.02 gives $|L-L^{\prime }|\le |L-a_N|+|a_N-L^{\prime }|<\epsilon +\epsilon =|L-L^{\prime }|$, a contradiction. So $L=L^{\prime }$. $\square$

Theorem (algebraic limit theorem). If $a_n\to a$ and $b_n\to b$, then: (i) $a_n+b_n\to a+b$; (ii) $c\cdot a_n\to c\cdot a$ for any $c\in \mathbb{R}$; (iii) $a_n{b}_n\to ab$; (iv) if $b\ne 0$ and $b_n\ne 0$ for all $n$, then $a_n/b_n\to a/b$.

Proof of (i). Fix $\epsilon >0$. Choose $N_1$ so $n\ge N_1$ gives $|a_n-a|<\epsilon /2$. Choose $N_2$ so $n\ge N_2$ gives $|b_n-b|<\epsilon /2$. Set $N=\max (N_1,N_2)$. For $n\ge N$, $|(a_n+b_n)-(a+b)|\le |a_n-a|+|b_n-b|<\epsilon /2+\epsilon /2=\epsilon$.

Proof of (ii). If $c=0$ the claim reads $0\to 0$. If $c\ne 0$, fix $\epsilon >0$ and choose $N$ so $n\ge N$ gives $|a_n-a|<\epsilon /|c|$. Then $|c{a}_n-ca|=|c|\cdot |a_n-a|<\epsilon$.

Proof of (iii). Decompose $a_n{b}_n-ab=(a_n-a)b+a_n(b_n-b)$. Because $a_n\to a$, the sequence $(a_n)$ is bounded: some $M>0$ has $|a_n|\le M$ for every $n$ (a convergent sequence is bounded, proved in the Full proof set). Set $B=\max (|b|,M,1)$. Fix $\epsilon >0$ and choose $N$ so $n\ge N$ gives both $|a_n-a|<\epsilon /(2B)$ and $|b_n-b|<\epsilon /(2B)$. Then $|a_n{b}_n-ab|\le |b|\cdot |a_n-a|+|a_n|\cdot |b_n-b|\le B\cdot \epsilon /(2B)+B\cdot \epsilon /(2B)=\epsilon$.

Proof of (iv). It is enough to show $1/b_n\to 1/b$; the quotient rule then follows from (iii). Since $b_n\to b\ne 0$, eventually $|b_n-b|<|b|/2$, so for large $n$, $|b_n|>|b|/2$. Fix $\epsilon >0$ and choose $N$ so $n\ge N$ gives $|b_n-b|<\epsilon b^2/2$ and $|b_n|>|b|/2$. Then $|1/b_n-1/b|=|b-b_n|/(|b_n|\cdot |b|)<(\epsilon b^2/2)/(b^2/2)=\epsilon$. $\square$

Theorem (squeeze theorem). If $a_n\le c_n\le b_n$ for every $n$ and $a_n\to L$, $b_n\to L$, then $c_n\to L$.

Proof. Fix $\epsilon >0$. Choose $N$ so $n\ge N$ gives both $|a_n-L|<\epsilon$ and $|b_n-L|<\epsilon$, i.e., $L-\epsilon <a_n$ and $b_n<L+\epsilon$. Sandwiching, $L-\epsilon <a_n\le c_n\le b_n<L+\epsilon$, so $|c_n-L|<\epsilon$. $\square$

Theorem (monotone convergence theorem). Every bounded monotone sequence of real numbers converges. A bounded-above monotone-increasing sequence converges to $\sup \{a_n:n\in \mathbb{N}\}$; a bounded-below monotone-decreasing sequence converges to $\inf \{a_n:n\in \mathbb{N}\}$.

Proof. Let $(a_n)$ be monotone increasing and bounded above. The set $S=\{a_n:n\in \mathbb{N}\}$ is non-empty and bounded above, so by the LUB axiom of 02.02.01 the supremum $L=\sup S$ exists in $\mathbb{R}$.

Fix $\epsilon >0$. Because $L$ is the least upper bound, $L-\epsilon$ is not an upper bound of $S$: some $a_N\in S$ has $a_N>L-\epsilon$. By monotonicity, for every $n\ge N$, $a_n\ge a_N>L-\epsilon$. Because $L$ is an upper bound, $a_n\le L<L+\epsilon$. Combining, $|a_n-L|<\epsilon$ for every $n\ge N$. So $a_n\to L$.

The monotone-decreasing case applies the same argument to $(-a_n)$, or directly to the greatest lower bound. $\square$

Theorem (subsequence characterization). A sequence $(a_n)$ converges to $L$ iff every subsequence of $(a_n)$ also converges to $L$.

Proof. ($\Rightarrow$) If $a_n\to L$, fix $\epsilon >0$ and choose $N$ so $n\ge N$ gives $|a_n-L|<\epsilon$. For any subsequence $(a_{n_k})$, the index map $k\mapsto n_k$ is strictly increasing, so $n_k\ge k$; thus for $k\ge N$, $n_k\ge N$ and $|a_{n_k}-L|<\epsilon$. ($\Leftarrow$) The full sequence is a subsequence of itself. $\square$

Corollary. If $(a_n)$ has two subsequences converging to different limits, then $(a_n)$ diverges.

Theorem (Bolzano-Weierstrass, statement). Every bounded sequence in $\mathbb{R}$ has a convergent subsequence. The full proof — via interval bisection producing a nested sequence of closed intervals whose lengths shrink to zero — appears in 02.03.02, where the theorem is paired with the Cauchy criterion and shown to be one of the equivalent reformulations of the completeness axiom of 02.02.01.

Bridge. The ε–$N$ definition of convergence builds toward 02.03.02 (where the Cauchy criterion and Bolzano-Weierstrass theorem are proved, and convergence is characterised without naming the limit) and appears again in 02.03.03 (where an infinite series is declared convergent exactly when its partial-sum sequence satisfies this definition). The foundational reason the definition is set up this way — naming the limit $L$ first, then producing an index $N$ for each tolerance $\epsilon$ — is that it makes the limit laws and the squeeze theorem direct consequences of the triangle inequality of 00.01.02. This is exactly the structural fact that lets later analysis inherit the same toolkit: a function-limit statement reduces to a sequence-limit statement by testing $f$ along sequences $x_n\to x$, and continuity in 02.04.02 becomes "preserve convergent sequences". The central insight is that convergence is preserved under the field operations (the algebraic limit theorem) and under order-sandwiching (the squeeze theorem), and these two preservation principles cover almost every concrete limit computation. Putting these together, the limit laws package the field structure of $\mathbb{R}$ as a calculus of limits, and the monotone convergence theorem packages the order structure of $\mathbb{R}$ (via the LUB axiom) as a limit-existence theorem; the two together form the working foundation on which 02.03.02 and 02.03.03 build.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: none — the ε–$N$ definition, the algebraic limit theorem, the squeeze theorem, and the monotone convergence theorem all have Mathlib counterparts. Concretely, Metric.tendsto_atTop unpacks the ε–$N$ definition on ℝ; Tendsto.add, Tendsto.mul, Tendsto.const_smul, and Tendsto.div supply the four parts of the algebraic limit theorem; the squeeze_succ lemma family supplies the squeeze theorem; and exists_tendsto_of_monotone_bounded together with the isLUB / csupr machinery supplies the monotone convergence theorem wired to the LUB structure of 02.02.01.

What Mathlib does not package in a single place is the undergraduate-curriculum form of these results side by side: the algebraic limit theorem collected as one named block, the squeeze theorem stated for sequences (rather than filters), and the standard worked estimates — the $1/n\to 0$ verification, the $(2n+1)/(3n-7)$ limit, the Cesàro mean theorem — gathered for pedagogy. The Codex namespace records this as a packaging task rather than a missing theorem.

Advanced results Master

The ε–$N$ definition supports two refinements that turn "does the sequence converge?" into a quantitative theory: the order-theoretic refinement (limsup and liminf) and the subsequence refinement (the set of subsequential limits). Both are grounded in the monotone convergence theorem and lead into the Cauchy and Bolzano-Weierstrass theory of 02.03.02.

Definition (limit superior and limit inferior). For a bounded sequence $(a_n)$, set $u_n={\sup }_{k\ge n}{a}_k$ and ${\ell }_n={\inf }_{k\ge n}{a}_k$. The sequence $(u_n)$ is monotone decreasing and bounded below; $({\ell }_n)$ is monotone increasing and bounded above. Both converge by the monotone convergence theorem. Define ${\mathrm{limsup}}_{n\to \infty }{a}_n={\lim }_{n\to \infty }{u}_n={\inf }_n{\sup }_{k\ge n}{a}_k$ and ${\mathrm{liminf}}_{n\to \infty }{a}_n={\lim }_{n\to \infty }{\ell }_n={\sup }_n{\inf }_{k\ge n}{a}_k$. Both exist as real numbers by the completeness axiom of 02.02.01.

Theorem (limsup/liminf characterization). Let $(a_n)$ be a bounded sequence. Then (i) $\mathrm{liminf}{a}_n\le \mathrm{limsup}{a}_n$; (ii) $(a_n)$ converges iff $\mathrm{liminf}{a}_n=\mathrm{limsup}{a}_n$, in which case $\lim a_n$ equals their common value; (iii) $\mathrm{limsup}{a}_n$ is the largest subsequential limit of $(a_n)$ and $\mathrm{liminf}{a}_n$ is the smallest.

Proof sketch. (i) For every $n$, ${\ell }_n\le u_n$; taking limits preserves the non-strict inequality by Exercise 6. (ii) If $a_n\to L$, then ${\ell }_n\le a_n\le u_n$ for every $n$ and both ${\ell }_n,u_n$ are squeezed to $L$, so $\mathrm{liminf}=\mathrm{limsup}=L$. Conversely, if $\mathrm{liminf}=\mathrm{limsup}=L$, then ${\ell }_n\le a_n\le u_n$ with both bounds converging to $L$, and the squeeze theorem gives $a_n\to L$. (iii) is a refinement of Bolzano-Weierstrass: extract a subsequence converging to $\mathrm{limsup}{a}_n$ by choosing indices $n_k$ with $a_{n_k}$ close to $u_{n_k}$; the full construction appears in 02.03.02. ^{[Royden Ch. 2]}

Theorem (subsequential limits form a closed set). The set $E$ of subsequential limits of a bounded sequence $(a_n)$ in $\mathbb{R}$ is a non-empty closed subset of $\mathbb{R}$. Its maximum is $\mathrm{limsup}{a}_n$ and its minimum is $\mathrm{liminf}{a}_n$.

Bolzano-Weierstrass (02.03.02) gives $E\ne \varnothing$. The order-completeness of $\mathbb{R}$ gives $\sup E\in E$ and $\inf E\in E$ by extracting subsequences whose terms approach the supremum and infimum. Closure under limits of sequences in $E$ is a diagonal argument: a sequence of subsequential limits, each realised by a subsequence of $(a_n)$, has a limit realised by a diagonal subsequence. The details belong with the compactness theory of 02.03.02.

Theorem (completeness equivalences, summary). Over the Archimedean ordered-field axioms of 02.02.01, the following are equivalent: (a) the LUB axiom; (b) the monotone convergence theorem; (c) the nested-interval property; (d) the Cauchy criterion (every Cauchy sequence converges); (e) the Bolzano-Weierstrass theorem.

The monotone convergence theorem implies the LUB axiom by approximating $\sup S$ with a monotone-increasing sequence of elements of $S$; the LUB axiom implies monotone convergence by the proof above; the nested-interval property is an equivalent geometric form; the Cauchy criterion's equivalence with convergence is proved in 02.03.02; and the Bolzano-Weierstrass theorem is shown there to be derivable from any of the other four. The five reformulations are the structural content of completeness on the real line. ^{[Rudin Ch. 3] [Abbott Ch. 2–3]}

Theorem (convergence in general metric spaces). The ε–$N$ definition specialises $\mathbb{R}$ to the case where the distance is $d(x,y)=|x-y|$. In a general metric space $(X,d)$ from 02.01.05, a sequence $(x_n)$ converges to $L\in X$ iff for every $\epsilon >0$ some $N$ has $d(x_n,L)<\epsilon$ for $n\ge N$.

The algebraic limit theorem does not generalise to metric spaces (it relies on the field structure of $\mathbb{R}$), but the squeeze theorem generalises in the form: if $d(x_n,L)\le d(y_n,L)$ and $y_n\to L$, then $x_n\to L$. The monotone convergence theorem relies essentially on the order structure of $\mathbb{R}$ and has no metric-space analogue. The Cauchy criterion, by contrast, generalises verbatim and becomes the working definition of completeness for metric spaces, leading to the Banach-space theory of 02.11.04. ^{[Dieudonné Ch. 3]}

Synthesis. The convergence theory of real sequences identifies three faces of limit existence on $\mathbb{R}$: the algebraic face (the limit laws), the order-theoretic face (the monotone convergence theorem together with limsup and liminf), and the metric face (the ε–$N$ definition itself, and the Cauchy criterion of 02.03.02). The foundational reason the three faces cohere is the completeness axiom of 02.02.01: each face reformulates completeness, and each reformulation generalises to a different broader setting — the algebraic face to topological groups and vector spaces, the order-theoretic face to complete lattices and ordered sets, the metric face to complete metric spaces and uniform spaces. This is exactly the structural decomposition that explains why calculus works on $\mathbb{R}$ but fails on $\mathbb{Q}$: the field operations are identical on both, but only $\mathbb{R}$ carries the completeness that turns bounded monotone lists into convergent lists. Putting these together identifies the algebraic limit theorem as the bridge from the field axioms to computable limits, the squeeze theorem as the bridge from the order axioms to inherited convergence, and the monotone convergence theorem as the bridge from the LUB axiom to limit existence. The bridge is the equivalence of the ε–$N$ definition, the Cauchy criterion, the monotone convergence theorem, and the LUB axiom — four formulations of one structural fact about $\mathbb{R}$, each natural in a different downstream setting.

Full proof set Master

Proposition (uniqueness of limits). Proved in §"Key theorem with proof" above.

Proposition (algebraic limit theorem, product rule, full estimate). If $a_n\to a$ and $b_n\to b$, then $a_n{b}_n\to ab$.

Proof. Decompose $a_n{b}_n-ab=(a_n-a)b+a_n(b_n-b)$. Because $a_n\to a$, the sequence $(a_n)$ is bounded: choose $N_0$ so $n\ge N_0$ gives $|a_n-a|<1$, and set $M_a=\max (|a_1|,\dots ,|a_{N_0}|,|a|+1)$, so $|a_n|\le M_a$ for every $n$. Set $B=\max (M_a,|b|,1)$. Fix $\epsilon >0$ and choose $N\ge N_0$ so $n\ge N$ gives both $|a_n-a|<\epsilon /(2B)$ and $|b_n-b|<\epsilon /(2B)$. Then

$$|a_n{b}_n-ab|\le |b|\cdot |a_n-a|+|a_n|\cdot |b_n-b|\le B\cdot \frac{\epsilon }{2B}+B\cdot \frac{\epsilon }{2B}=\epsilon .\square$$

Proposition (convergent sequences are bounded). If $a_n\to a$, then $(a_n)$ is bounded.

Proof. Take $\epsilon =1$ in the convergence definition: some $N$ has $|a_n-a|<1$ for $n\ge N$, so $|a_n|\le |a|+1$ for $n\ge N$. Set $M=\max (|a_1|,\dots ,|a_{N-1}|,|a|+1)$; then $|a_n|\le M$ for every $n$. $\square$

Proposition (squeeze theorem). Proved in §"Key theorem with proof" above.

Proposition (monotone convergence theorem). Proved in §"Key theorem with proof" above.

Proposition (preservation of non-strict inequalities). If $a_n\to a$, $b_n\to b$, and $a_n\le b_n$ for every $n$, then $a\le b$.

Proof. Suppose $a>b$. Set $\epsilon =(a-b)/2>0$. Choose $N$ so $n\ge N$ gives $|a_n-a|<\epsilon$ and $|b_n-b|<\epsilon$. Then $a_n>a-\epsilon =(a+b)/2$ and $b_n<b+\epsilon =(a+b)/2$, so $a_n>b_n$, contradicting the hypothesis. So $a\le b$. $\square$

Proposition (Cesàro mean theorem). If $a_n\to a$, then $b_n=(a_1+\cdots +a_n)/n\to a$.

Proof. Given $\epsilon >0$, choose $N_0$ so $n\ge N_0$ implies $|a_n-a|<\epsilon /2$. Set $C={\sum }_{k=1}^{N_0}|a_k-a|$, a finite constant. Then

$$|b_n-a|=\left|\frac{1}{n}\sum _{k=1}^n(a_k-a)\right|\le \frac{1}{n}\sum _{k=1}^{N_0}|a_k-a|+\frac{1}{n}\sum _{k=N_0+1}^n|a_k-a|\le \frac{C}{n}+\frac{n-N_0}{n}\cdot \frac{\epsilon }{2}.$$

Choose $N\ge \max (N_0,\lceil 2C/\epsilon \rceil )$. For $n\ge N$, $C/n\le \epsilon /2$ and $(n-N_0)/n<1$, so $|b_n-a|<\epsilon /2+\epsilon /2=\epsilon$. $\square$

The converse fails: $a_n=(-1{)}^n$ diverges but its Cesàro means converge to $0$. Cesàro summability is therefore strictly weaker than ordinary convergence, a fact used in the summability theory of Fourier series.

Proposition (subsequence characterization). $a_n\to L$ iff every subsequence of $(a_n)$ converges to $L$.

Proof. ($\Rightarrow$) If $a_n\to L$, fix $\epsilon >0$ and choose $N$ so $n\ge N$ gives $|a_n-L|<\epsilon$. For any subsequence $(a_{n_k})$, strict increase of $k\mapsto n_k$ gives $n_k\ge k$, so $k\ge N$ implies $n_k\ge N$ and $|a_{n_k}-L|<\epsilon$. ($\Leftarrow$) The full sequence is a subsequence of itself. $\square$

Connections Master

• Real-number axioms 02.02.01. The ε–$N$ definition of sequence convergence is the sequential translation of the LUB completeness axiom: every bounded monotone sequence converges because the supremum of its range exists as a real number. The monotone convergence theorem proved in this unit is one of the equivalent reformulations of completeness identified in 02.02.01 and shown equivalent to the Cauchy criterion in 02.03.02. The Archimedean property of 02.02.01 is the explicit engine behind every concrete ε–$N$ estimate, including the $1/n\to 0$ computation in the worked example and the $(3n+1)/(n+2)\to 3$ argument of Exercise 3.

• Cauchy sequences and Bolzano-Weierstrass 02.03.02. The Cauchy criterion — convergence characterised without naming the limit — is the metric reformulation of the ε–$N$ definition developed here. The Bolzano-Weierstrass theorem — every bounded sequence has a convergent subsequence — is the compactness reformulation. Both are proved in 02.03.02, and their equivalence with the monotone convergence theorem of this unit is the structural content of the completeness axiom. The set of subsequential limits studied in the Advanced results above is closed and non-empty by Bolzano-Weierstrass, with $\mathrm{limsup}$ and $\mathrm{liminf}$ as its extreme points.

• Infinite series 02.03.03. An infinite series $\sum a_n$ converges iff its sequence of partial sums $s_N=a_1+\cdots +a_N$ converges in the sense of this unit. Every convergence test in 02.03.03 — comparison, ratio, alternating — ultimately reduces to checking that the partial-sum sequence satisfies the ε–$N$ definition, usually via the Cauchy criterion of 02.03.02 or the monotone convergence theorem of this unit. The limit laws proved here are what license termwise algebraic manipulation of convergent series.

• Continuous functions 02.04.02. A function $f:\mathbb{R}\to \mathbb{R}$ is continuous at $x$ iff for every sequence $x_n\to x$, the image sequence $f(x_n)\to f(x)$. This sequential characterization reduces function limits to sequence limits, and every theorem about continuous functions (the intermediate value theorem, the extreme value theorem, uniform continuity) is proved by composing sequences with $f$ and applying the limit laws and the squeeze theorem of this unit.

• Metric spaces 02.01.05. The ε–$N$ definition generalises from $\mathbb{R}$ with absolute value to any metric space by replacing $|x-y|$ with $d(x,y)$. The algebraic limit theorem does not generalise (it needs field structure), but the squeeze theorem and the Cauchy criterion do. Metric-space convergence is the natural language for the Banach-space and Hilbert-space theory of 02.11.04, where the Cauchy criterion becomes the defining property of completeness.

Historical & philosophical context Master

Augustin-Louis Cauchy's Cours d'analyse of 1821 ^{[Cauchy 1821]} gave the first systematic treatment of sequence convergence. Cauchy wrote that a sequence $(a_n)$ converges to $L$ when the differences $|a_n-L|$ become "as small as one wishes" for sufficiently large $n$ — the informal statement of the ε–$N$ definition. Cauchy used this definition to prove the algebraic limit theorem, the squeeze principle, and the convergence of geometric series, establishing the working toolkit that this unit formalises. Cauchy did not separate the tolerance from the cutoff symbolically; that notational step came later.

Bernard Bolzano's 1817 Rein analytischer Beweis ^{[Bolzano 1817]}, written four years before Cauchy's Cours, contained the bounded-monotone convergence argument as a lemma embedded in the first analytic proof of the intermediate value theorem. Bolzano recognised that a bounded set of real numbers has a least upper bound (his formulation of completeness) and used it to extract convergent subsequences; the monotone convergence theorem of this unit is a direct descendant of Bolzano's lemma. The paper was largely unread outside Prague for half a century.

Karl Weierstrass's Berlin lectures of the 1860s ^{[Weierstrass 1894]} standardised the ε–$N$ (and ε–$\delta$) notation that made the limit definition fully rigorous. The notation separates the two quantifiers — "for every $\epsilon$" and "there exists an $N$" — that Cauchy had run together, and is the form in which the definition is taught today. Weierstrass's rigorisation is what allowed the late-nineteenth-century unification of sequence convergence, continuous functions, and Riemann integration under a single ε-based framework.

The modern textbook presentation in Rudin's Principles of Mathematical Analysis Ch. 3 ^{[Rudin Ch. 3]} and Abbott's Understanding Analysis Ch. 2–3 ^{[Abbott Ch. 2–3]} arranges the results in the order: ε–$N$ definition, algebraic limit theorem, squeeze theorem, monotone convergence theorem, subsequences, then Bolzano-Weierstrass and the Cauchy criterion (here deferred to 02.03.02). This unit follows that arrangement, with Dieudonné's filter-based treatment ^{[Dieudonné Ch. 3]} and Royden's limsup/liminf presentation ^{[Royden Ch. 2]} supplying the Master-tier generalisations.

Bibliography Master

@book{Cauchy1821Cours,
  author    = {Cauchy, Augustin-Louis},
  title     = {Cours d'analyse de l'{\'E}cole royale polytechnique. 1.re partie. Analyse alg{\'e}brique},
  publisher = {Debure fr{\`e}res},
  address   = {Paris},
  year      = {1821}
}

@book{Bolzano1817Rein,
  author    = {Bolzano, Bernard},
  title     = {Rein analytischer Beweis des Lehrsatzes, da{\ss} zwischen je zwey Werthen, die ein entgegengesetztes Resultat gew{\"a}hren, wenigstens eine reelle Wurzel der Gleichung liege},
  publisher = {Gottlieb Haase},
  address   = {Prag},
  year      = {1817}
}

@book{Weierstrass1894Werke,
  author    = {Weierstrass, Karl},
  title     = {Mathematische Werke},
  publisher = {Mayer \& M{\"u}ller},
  address   = {Berlin},
  year      = {1894}
}

@book{RudinPrinciples,
  author    = {Rudin, Walter},
  title     = {Principles of Mathematical Analysis},
  edition   = {3rd},
  publisher = {McGraw-Hill},
  year      = {1976}
}

@book{AbbottUnderstanding,
  author    = {Abbott, Stephen},
  title     = {Understanding Analysis},
  edition   = {2nd},
  publisher = {Springer},
  year      = {2015}
}

@book{DieudonneFoundations,
  author    = {Dieudonn{\'e}, Jean},
  title     = {Foundations of Modern Analysis},
  publisher = {Academic Press},
  year      = {1960}
}

@book{RoydenRealAnalysis,
  author    = {Royden, H. L.},
  title     = {Real Analysis},
  edition   = {3rd},
  publisher = {Macmillan},
  year      = {1988}
}