Notes in Modern Analysis

Lecture Notes

• Chapter 0Foundations: Logic, Sets, Functions, and the Craft of ProofCourse preliminariesOverviewThese preliminaries are the on-ramp to the course. They assume you have never written a proof and have never met set notation, and they build—slowly and from the ground up—everything the lectures take for granted: how to read and write mathematical statements (0.1), the quantifiers that power every definition in analysis, the language of sets, relations, and functions, and, at its centre, the craft of proof: how a proof is built, the handful of standard strategies (which are exactly the Modes of Argument tagged throughout this book), and how to find one when you are stuck. We close with a worked, classified problem set. Read these preliminaries actively, with pencil in hand; nothing here is meant to be watched go by.
• Lecture 1From the Naturals to the Reals: Induction, Ordered Fields, and CompletenessTuesday, 26 May 2026OverviewThe course opens with a deceptively simple question—is 0.999…=1?—whose honest answer requires building the real numbers from the ground up. We begin with the natural numbers and induction, construct ℤ and ℚ, and isolate the field and order axioms they satisfy. The rationals then fall short for analysis: √2 is missing, and the sets bounding it have neither a largest nor a smallest element. Repairing that gap is the least-upper-bound property, which characterises ℝ; from it follow the Archimedean property and the density of ℚ—and, at last, the resolution of 0.999…=1. The companion text is Rudin, Chapter 1, especially the ordered-field and least-upper-bound material r 1.5–r 1.20.
• Lecture 2Counting the Infinite, and the Geometry of Distance: Countability and Metric SpacesThursday, 28 May 2026OverviewLecture 1 built ℝ as the complete ordered field and drew from completeness the Archimedean property (1.7.1) and the density of ℚ (1.7.2); the same property gives infima as readily as suprema (1.5.3). Two new questions open here. First, how large is ℝ? Cantor's notion of countability lets us compare infinite sets, and his diagonal argument shows ℝ is strictly larger than ℚ—uncountable, with ℚ a vanishingly “thin” subset. Second, what is the right language for distance? Metric spaces distil the Euclidean norm into three axioms, and the open sets they generate are the gateway to topology, where the rest of the course lives. The companion text is Rudin, Chapter 2, especially r 2.1–r 2.19.
• Lecture 3The Topology of Metric Spaces: Open and Closed Sets, and CompactnessTuesday, 2 June 2026OverviewLecture 2 introduced metric spaces and the open ball (2.3.1). This lecture builds the topology they generate: which sets are open—closed under arbitrary unions and finite intersections—then the limit (cluster) points that define closed sets, the clean duality E open ⟺ E^c closed, how openness is relative to a subspace, and finally compactness: the open-cover/finite-subcover property that, unlike open or closed, is intrinsic to a set. One method recurs throughout: to show a set is open, write it as a union of balls centred at its own points. The companion text is Rudin, Chapter 2, especially r 2.18–r 2.33.
• Lecture 4Compactness: Heine–Borel and Bolzano–WeierstrassThursday, 4 June 2026OverviewLecture 3 defined compactness—every open cover has a finite subcover (3.4.1)—and showed it is intrinsic (3.4.4). Here we draw out its consequences: a compact set is closed and bounded, a closed subset of a compact set is compact, and nested nonempty compact sets meet. Specialising to ℝ^k through nested intervals and k-cells yields the two pillars of the subject—Heine–Borel (in ℝ^k, compact = closed and bounded) and Bolzano–Weierstrass (every bounded infinite set has a cluster point). A short topological coda—interior, closure, connectedness—closes the chapter.
• Lecture 5Sequences: Convergence, Cauchy Completeness, and ℝ^kTuesday, 9 June 2026OverviewA sequence converges to p when it eventually lies in every ball about p. Two things must hold: the terms must settle, and a limit must exist in the space—the second can fail. We develop the calculus of limits (the open-set view, uniqueness, boundedness, the algebra of limits in ℝ/ℂ, coordinatewise convergence in ℝ^k), then subsequences—every sequence in a compact space has a convergent one—and finally Cauchy sequences and completeness: a Cauchy sequence is one whose terms bunch together, every convergent sequence is Cauchy, and a space is complete when the converse holds. Compact spaces and ℝ^k are complete.
• Lecture 6Monotone Convergence, the Squeeze Theorem, and the Upper and Lower LimitsThursday, 11 June 2026OverviewPicking up from completeness, this lecture assembles the basic machinery for limits of real sequences. Monotonicity plus boundedness already forces convergence; the comparison and squeeze principles let limits be read off from neighbours; and for sequences that oscillate, the upper and lower limits limsup and liminf record the band the sequence is eventually trapped in, coinciding exactly when the sequence converges. After recording four limits that recur throughout the subject, we open the theory of series: convergence through partial sums, the Cauchy criterion, and the first divergence test. The companion text is Rudin, Chapter 3, especially r 3.12–r 3.40.
• Lecture 7Function Limits, Continuity, and the Topology of Continuous MapsThursday, 18 June 2026OverviewThis lecture begins Chapter 4 by replacing the sequential limit p_n → p with the function limit lim_x→pf(x)=q in metric spaces. The ε–δ definition is tied back to sequences, then specialized to continuity at a point and continuity on a set. With the open-set characterization in hand, continuity becomes a topological operation: preimages of open sets are open, compositions are continuous, compact sets have compact images, connected sets have connected images, and the classical extreme-value and intermediate-value theorems become consequences. The lecture closes with homeomorphisms and the compactness theorem that upgrades pointwise continuity to uniform continuity. The companion text is Rudin, Chapter 4, especially r 4.1–r 4.26.
• Lecture 8Derivatives, Taylor’s Theorem, and AnalyticityTuesday, 23 June 2026OverviewThis lecture begins the differential part of the course. The derivative is first defined by the difference quotient, then immediately reinterpreted as the best first-order linear approximation. That viewpoint makes the usual rules of differentiation bookkeeping in powers of the increment, and it also explains the multivariable definition. The lecture then turns the derivative into a global tool through Fermat's interior-extremum theorem, Rolle's theorem, the Mean Value Theorem, monotonicity, and Darboux's theorem. It closes with Taylor's theorem and the distinction between smooth and analytic functions, first over ℝ and then over ℂ. The companion text is Rudin, Chapter 5, with the closest matches marked by the locators r 5.1–r 5.16.
• Lecture 9The Riemann Integral: Construction, Integrability, and the Fundamental TheoremThursday, 25 June 2026OverviewThe calculus slogan is “area under the curve,” but here the integral comes first: area is defined by the integral, not the reverse. We build the Riemann integral for any bounded f on a closed bounded interval—continuity is not required—by trapping the curve between under- and over-estimating staircases, and call f integrable exactly when the two estimates can be made to agree. Refining a partition squeezes the sums together, and that single fact yields the working test for integrability: U−L can be made smaller than any ε. The test then delivers three classes of integrable functions—continuous, monotone, and boundedly discontinuous at finitely many points—and points past Ch. 6 to the full Lebesgue criterion, which the Cantor set shows is genuinely sharper. We close by recording the algebraic properties of the integral and the two halves of the Fundamental Theorem, whose proofs open the next lecture. The companion text is Rudin, Chapter 6, read throughout with α(x)=x, so ∫f dα becomes ∫f dx (r 6.1–r 6.21).

Problem Sets

• Homework 1The Real NumbersRudin, Chapters 1–2In this setInduction · an equivalence relation · an ordered-field obstruction — with worked solutions and reflections.
• Homework 2The Topology of Metric SpacesRudin, Chapter 2In this setLimit points · compactness · perfect, connected & separable sets — with worked solutions and reflections.
• Homework 3Sequences, Series, and ContinuityRudin, Chapters 3–4In this setSequences · series · continuity — with worked solutions and reflections.

Resources

• RudinPrinciples of Mathematical AnalysisPrimary course text · 3rd ed. · LionMailAboutThe text the course follows; the lectures track it chapter by chapter, and the notes’ Rudin locators (r 1.5, …) point straight into it. Chapters 1–7. Opens in the Columbia-only Drive — sign in with LionMail.
• PughReal Mathematical AnalysisCourse companion · 2nd ed., Springer · LionMailAboutA geometric, picture-driven companion to Rudin — especially strong on the construction of ℝ, metric spaces, and the Cantor set.
• AbbottUnderstanding AnalysisSupplementary · 2nd ed., Springer · LionMailAboutA gentler, motivation-first analysis text — excellent for building intuition before Rudin’s terseness.
• VellemanHow to Prove It: A Structured ApproachProof technique · 3rd ed., Cambridge · LionMailAboutThe standard reference for writing proofs — logic, quantifiers, sets, induction, and how to structure an argument.
• CummingsProofs: A Long-Form Mathematics TextbookProof technique · free from the authorAboutA friendly, conversational introduction to proof-writing — free PDF from the author at longformmath.com.
• CummingsReal Analysis: A Long-Form Mathematics TextbookSupplementary · free from the authorAboutA long-form, intuition-rich analysis text — free PDF from the author at longformmath.com.
• LeblBasic Analysis: Introduction to Real AnalysisSupplementary · free, open-licensedAboutA free, Creative-Commons real analysis textbook — a clean, complete reference you can read or download anytime.
• YuA Complete Solution Guide to Principles of Mathematical AnalysisSolutions to Rudin · LionMailAboutWorked solutions to Rudin’s exercises — for checking your work after you’ve genuinely tried them yourself.