Elementary Analysis

An interactive companion to Kenneth A. Ross, Elementary Analysis: The Theory of Calculus, 2nd ed. (Springer, 2013). Section numbers (§7, Definition 7.1, Exercise 7.3, …) match the book, so you can read the two side by side.

Each section page has definitions and theorems in coloured boxes, proofs you can expand, interactive figures, and exercises with hints and solutions you can reveal. Each section ends with a short self-check quiz. Citations such as Theorem 9.1 are links: hover over or focus one to see the statement. You can mark exercises done and sections read; your progress is saved in this browser.

Sections marked ★ are the book’s optional “enrichment” sections. ✓ means the page is written.

Alongside Ross’s text, each page has short companion notes that are not in the book: Why this matters (motivation), Intuition, Common pitfall, and Reading alongside Abbott, which points to where the same ideas appear in Stephen Abbott’s Understanding Analysis (2nd ed., Springer, 2015) and how the two books’ approaches differ. The notes are written for this site; they do not reproduce either book.

Contents

Chapter	Sections
1 · Introduction	§1 The Set ℕ of Natural Numbers ✓ · §2 The Set ℚ of Rational Numbers ✓ · §3 The Set ℝ of Real Numbers ✓ · §4 The Completeness Axiom ✓ · §5 The Symbols +∞ and −∞ ✓ · §6 ★ A Development of ℝ ✓
2 · Sequences	§7 Limits of Sequences ✓ · §8 A Discussion about Proofs ✓ · §9 Limit Theorems for Sequences ✓ · §10 Monotone Sequences and Cauchy Sequences ✓ · §11 Subsequences ✓ · §12 lim sup’s and lim inf’s ✓ · §13 ★ Some Topological Concepts in Metric Spaces ✓ · §14 Series ✓ · §15 Alternating Series and Integral Tests ✓ · §16 ★ Decimal Expansions of Real Numbers ✓
3 · Continuity	§17 Continuous Functions ✓ · §18 Properties of Continuous Functions ✓ · §19 Uniform Continuity · §20 Limits of Functions · §21★ More on Metric Spaces: Continuity · §22★ More on Metric Spaces: Connectedness
4 · Sequences and Series of Functions	§23 Power Series · §24 Uniform Convergence · §25 More on Uniform Convergence · §26 Differentiation and Integration of Power Series · §27★ Weierstrass’s Approximation Theorem
5 · Differentiation	§28 Basic Properties of the Derivative · §29 The Mean Value Theorem · §30★ L’Hospital’s Rule · §31 Taylor’s Theorem
6 · Integration	§32 The Riemann Integral · §33 Properties of the Riemann Integral · §34 Fundamental Theorem of Calculus · §35★ Riemann–Stieltjes Integrals · §36★ Improper Integrals
7 · Capstone	§37★ A Discussion of Exponents and Logarithms · §38★ Continuous Nowhere-Differentiable Functions

Reading alongside Abbott

Abbott’s Understanding Analysis covers much of the same ground with more discussion and motivation, so it makes a good second reading. Abbott’s section numbers are written out as “Section 2.5” to keep them apart from Ross’s §-numbers.

Ross (this site)	Abbott, Understanding Analysis	How the approaches differ
§1 Natural numbers	Section 1.2 (induction)	Abbott takes ℕ as known; no Peano axioms.
§2 Rational numbers	Section 1.1	Abbott proves √2 irrational by the classical even/odd argument; Ross’s Rational Zeros Theorem generalizes it.
§3 Real numbers	Sections 1.2, 1.3	Abbott assumes an ordered field instead of listing the axioms.
§4 Completeness Axiom	Sections 1.3, 1.4	Abbott derives the nested interval property first and uses it often.
§5 The symbols ±∞	none	Abbott treats ±∞ informally.
§6 ★ A development of ℝ	Section 8.6	Same construction by Dedekind cuts.
§7–§8 Limits and proofs	Section 2.2	Same definition, also phrased with neighbourhoods.
§9 Limit theorems	Section 2.3	Algebraic and Order Limit Theorems.
§10 Monotone and Cauchy sequences	Sections 2.4, 2.6	Abbott proves the Cauchy criterion from Bolzano–Weierstrass; Ross uses lim sup and lim inf.
§11 Subsequences	Section 2.5	Abbott proves Bolzano–Weierstrass by repeated halving.
§12 lim sup and lim inf	Exercise 2.4.7	Treated only in an exercise.
§13 ★ Metric space topology	Sections 3.1–3.3, 8.2	Abbott works in ℝ and defines compactness by sequences.
§14 Series	Sections 2.4, 2.7, 8.3	Abbott adds the Cauchy condensation test; the Ratio Test is an exercise.
§15 Alternating and integral tests	Sections 2.1, 2.7	No integral test; rearrangements are a central theme.
§16 ★ Decimal expansions	Section 1.6	Decimals appear in Cantor’s diagonal argument.
§17 Continuous functions	Sections 4.1–4.3	Abbott defines continuity by ε–δ and proves the sequence version.
§18 Properties of continuous functions	Sections 4.4, 4.5	Via compact and connected sets; two proofs of the IVT.