Sequences and Their Limits — Mathematics Knowledge System

Overview

A sequence is a function from the natural numbers to the real numbers, written a₁, a₂, a₃, ... or (aₙ). The limit of a sequence is the value its terms 'tend to' as n grows without bound — a sequence with a finite limit is called convergent; otherwise it diverges. The rigorous ε-N definition of sequential convergence is one of the two central limiting notions of analysis (alongside the ε-δ limit of a function), and the two are intimately connected: sequential limits give the cleanest route to defining continuity, series, and completeness, and are often called the fundamental notion on which the whole of mathematical analysis ultimately rests.

Intuition

Picture a sequence of numbers as dots plotted at heights a₁, a₂, a₃, ... above the integers 1, 2, 3, .... The sequence converges to L if, no matter how thin a horizontal band you draw around the height L, all the dots eventually enter that band and never leave it again — only finitely many dots (a fixed, though possibly large, number of initial terms) are allowed to lie outside. The index N is 'how far out you must go' before every subsequent term is trapped inside the band; a thinner band (smaller ε) generally forces you to go further out (larger N).

Interactive Graph

Formal Definition

Definition

A sequence (aₙ) of real numbers converges to a limit c ∈ ℝ, written aₙ → c or lim(n→∞) aₙ = c, if:

\forall \varepsilon > 0,\ \exists N \in \mathbb{N} \text{ such that } \forall n \ge N,\ |a_n - c| < \varepsilon

For every tolerance ε there is an index N beyond which every term of the sequence lies within ε of c

ε-N definition of sequential convergence

\lim_{n \to \infty} a_n = c

Standard limit notation for a convergent sequence

a_n = \frac{1}{n} \ \Rightarrow\ a_n \to 0,\quad \text{since for any } \varepsilon>0,\ N > \tfrac{1}{\varepsilon} \text{ guarantees } |a_n - 0| < \varepsilon \text{ for all } n \ge N

Canonical example: the harmonic sequence converges to 0

Notation

Notation	Meaning
$(a_n)_{n=1}^{\infty}$	A sequence indexed by the natural numbers, alternatively written {aₙ}
$a_n \to c$	The sequence (aₙ) converges to the limit c
$N \in \mathbb{N}$	A natural-number threshold index: the point beyond which all terms are within ε of the limit
$\limsup_{n\to\infty} a_n,\ \liminf_{n\to\infty} a_n$	Limit superior/inferior — the largest/smallest subsequential limit; exist even when the sequence itself diverges
$a_n \to \infty$	Divergence to infinity: for every M, terms eventually exceed M (a distinct notion from convergence)

Derivation

Proving directly from the ε-N definition that the sequence aₙ = 1/n converges to 0 — the standard model for all such proofs, exactly analogous to an ε-δ proof but with N replacing δ, and 'n ≥ N' replacing '0<|x-a|<δ'.

|a_n - 0| = \left|\frac{1}{n} - 0\right| = \frac{1}{n}

Simplify the target quantity

\frac{1}{n} < \varepsilon \iff n > \frac{1}{\varepsilon}

Solve the inequality for n to discover how large N must be

\text{Choose } N \text{ to be any natural number with } N > \frac{1}{\varepsilon} \text{ (e.g. } N = \lceil 1/\varepsilon \rceil + 1 \text{)}

By the Archimedean property of ℝ, such an N always exists

n \ge N > \frac{1}{\varepsilon} \implies \frac{1}{n} \le \frac{1}{N} < \varepsilon

Forward verification completes the proof

Proofs

lim(n→∞) 1/n = 0

$\text{Let } \varepsilon>0 \text{ be given.}$ (The proof must hold for an arbitrary positive ε)
$\text{By the Archimedean property of } \mathbb{R}, \exists N\in\mathbb{N} \text{ with } N > 1/\varepsilon.$ (The Archimedean property guarantees natural numbers exceed any real bound)
$\text{Suppose } n \ge N.$ (Assume the hypothesis to be verified)
$\left|\frac{1}{n}-0\right| = \frac{1}{n} \le \frac{1}{N} < \varepsilon$ (n≥N>0 implies 1/n ≤ 1/N (reversing inequality under reciprocation of positives); then 1/N<ε by choice of N)
$\therefore |a_n - 0| < \varepsilon \text{ whenever } n \ge N.$ (Exactly the ε-N condition, confirming the limit is 0)

Every convergent sequence is bounded

$\text{Suppose } a_n \to L.$ (Given)
$\text{Apply the definition with } \varepsilon=1: \exists N \text{ such that } n\ge N \implies |a_n - L| < 1.$ (Instantiate the ε-N definition at the specific value ε=1)
$\text{For } n \ge N: |a_n| \le |a_n - L| + |L| < 1 + |L|.$ (Triangle inequality)
$\text{Let } M = \max\{|a_1|,\ldots,|a_{N-1}|,\ 1+|L|\}.$ (Take the max over the finitely many 'early' terms and the tail bound)
$\text{Then } |a_n| \le M \text{ for all } n.$ (Every term, early or in the tail, is bounded by M, so the sequence is bounded)

Properties

Uniqueness of limits

\text{If } a_n \to L_1 \text{ and } a_n \to L_2, \text{ then } L_1 = L_2.

Condition: Proved by the same triangle-inequality contradiction argument used for function limits.

Algebra of limits

a_n \to A,\ b_n \to B \implies a_n+b_n \to A+B,\quad a_nb_n \to AB,\quad a_n/b_n \to A/B\ (B\neq 0)

Monotone Convergence Theorem

\text{Every bounded monotone sequence of reals converges.}

Condition: Relies on the completeness (least-upper-bound property) of ℝ; fails over ℚ, e.g. rational approximations to √2 are bounded, increasing, but have no rational limit.

Bolzano–Weierstrass Theorem

\text{Every bounded sequence in } \mathbb{R} \text{ has a convergent subsequence.}

Condition: A cornerstone compactness-flavored result; foundational for proving the extreme value theorem.

Cauchy criterion

(a_n) \text{ converges} \iff \forall \varepsilon>0\ \exists N: \forall m,n\ge N,\ |a_n-a_m|<\varepsilon

Condition: Allows proving convergence without knowing the limit in advance — the defining property of a complete metric space.

Applications

Iterative numerical algorithms (Newton's method, gradient descent, fixed-point iteration) are analyzed as sequences, with convergence proofs directly using the ε-N framework to bound how many iterations guarantee a desired precision.

Worked Examples

Solve 1/n < 0.01 for n.
$n > \frac{1}{0.01} = 100$

Answer: Any N ≥ 101 (e.g. N = 101)

Practice Problems

Difficulty 5/10

Find N (in terms of ε) proving aₙ = 3/n² → 0.

Difficulty 4/10

Which theorem guarantees that a bounded sequence of real numbers, even if not monotone, must have a convergent subsequence?

The Squeeze TheoremThe Bolzano–Weierstrass TheoremThe Cauchy CriterionThe Archimedean Property

Difficulty 7/10

Prove that if aₙ → L and L > 0, then there exists N such that aₙ > 0 for all n ≥ N.

Common Mistakes

Common Mistake

Believing that a sequence's terms must be monotone (always increasing or decreasing) in order to converge.

Convergence only requires terms to eventually cluster arbitrarily close to L — oscillation is allowed as long as it dampens out. E.g. aₙ = (-1)ⁿ/n oscillates in sign forever but converges to 0.

Common Mistake

Confusing 'bounded' with 'convergent' — assuming every bounded sequence converges.

Boundedness is necessary but not sufficient. aₙ = (-1)ⁿ is bounded (between -1 and 1) but never converges — it perpetually oscillates between two values with no single limit.

Common Mistake

Treating N as if it must be the smallest possible index, or unique.

Just as with δ in the ε-δ definition, N is never unique — any larger N also satisfies the condition. Proofs need only exhibit some valid N, not the minimal one.

Quiz

The ε-N definition of aₙ → L requires:

∃N ∀ε: n≥N ⟹ |aₙ-L|<ε∀ε>0 ∃N: n≥N ⟹ |aₙ-L|<ε∀ε>0 ∀N: n≥N ⟹ |aₙ-L|<ε∀n ∃ε: |aₙ-L|<ε

Which of these sequences is bounded but does NOT converge?

aₙ = 1/naₙ = (-1)ⁿaₙ = naₙ = 2 for all n

Flashcards

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Historical Background

The idea of a sequence approaching a fixed value is ancient — Archimedes' method of exhaustion (3rd century BCE) approximated areas via sequences of polygons — but it long lacked a rigorous definition. Cauchy's 1821 Cours d'Analyse introduced what is now called a Cauchy sequence and described convergence in the same semi-rigorous verbal style he used for function limits. Weierstrass's 1850s–60s Berlin lectures gave the modern symbolic ε-N definition, exactly mirroring his ε-δ definition for functions. Georg Cantor's work in the 1870s on the completeness of the real numbers (via Cauchy sequences, as an alternative to Dedekind cuts) cemented the sequence limit as foundational to the construction of ℝ itself.

c. 250 BCE
Archimedes uses sequences of inscribed and circumscribed polygons to bound the area of a circle (method of exhaustion)
Archimedes
1821
Cauchy describes sequence convergence and introduces the Cauchy criterion in Cours d'Analyse
Augustin-Louis Cauchy
1860s
Weierstrass formalizes the ε-N definition of sequential convergence in his Berlin lectures
Karl Weierstrass
1872
Cantor constructs the real numbers as equivalence classes of Cauchy sequences of rationals
Georg Cantor

Summary

A sequence (aₙ) converges to L if ∀ε>0 ∃N∈ℕ such that n≥N ⟹ |aₙ-L|<ε — the same quantifier structure as the ε-δ definition, with N replacing δ.
Limits of sequences are unique, and every convergent sequence is bounded (though the converse is false).
The Monotone Convergence Theorem (bounded + monotone ⟹ convergent) relies on the completeness of ℝ and fails over ℚ.
The Bolzano–Weierstrass Theorem guarantees every bounded sequence has a convergent subsequence, even if the full sequence diverges.
The Cauchy criterion characterizes convergence intrinsically (terms get arbitrarily close to each other) without needing to know the limit in advance.

References

WebsiteWikipedia — Limit of a sequence
BookRudin, W. Principles of Mathematical Analysis, 3rd ed. Ch. 3.
BookCauchy, A.-L. (1821). Cours d'Analyse de l'École Royale Polytechnique.