--- title: Real Analysis description: Real analysis is the rigorous study of real numbers, sequences, series, and continuous functions. You will understand the formal proofs behind calculus concepts, enabling you to construct logical mathematical arguments and analyze the foundational structure of mathematical analysis. category: mathematics subcategory: calculus difficulty: beginner, intermediate, advanced url: /subject/real-analysis --- # Real Analysis Real analysis is the rigorous study of real numbers, sequences, series, and continuous functions. You will understand the formal proofs behind calculus concepts, enabling you to construct logical mathematical arguments and analyze the foundational structure of mathematical analysis. ## Where to start Start with Stephen Abbott's Understanding Analysis, a lucid introduction that focuses on why calculus works. If you only use one resource, make it Casey Rodriguez's free Real Analysis (MIT 18.100A) from MIT OpenCourseWare, with 25 lecture videos, notes, problem sets and exams. Charles Chapman Pugh's Real Mathematical Analysis, a visual, modern alternative to Rudin, is the natural next step. ## Available Resources 9 Books • 8 Courses • 7 Websites ## Websites ### 1. Real Analysis (AoPS Wiki) The AoPS wiki overview of real analysis: completeness of the reals, convergence and divergence of sequences, limits, continuity, differentiation and integration, with links to related articles. A short orientation to the vocabulary before working through a textbook. **Difficulty:** Beginner | **Price:** Free **Link:** https://artofproblemsolving.com/wiki/index.php/Real_analysis **Tags:** real-analysis, sequences, convergence, limits, continuity ### 2. Pauls Online Math Notes Complete lecture-note sets and worked practice problems from Paul Dawkins at Lamar University, spanning algebra, trigonometry, and calculus I through III plus differential equations. Useful for working problems step by step when a textbook explanation stalls. **Difficulty:** Beginner | **Price:** Free **Link:** https://tutorial.math.lamar.edu/ **Tags:** algebra, calculus, differential-equations, lecture-notes, practice-problems ### 3. Khan Academy Math Free video lessons and adaptive exercise sets covering arithmetic through calculus, linear algebra and statistics, with mastery tracking that identifies gaps. Suited to rebuilding school mathematics systematically before attempting university-level courses. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.khanacademy.org/math **Tags:** arithmetic, algebra, calculus, statistics, video-lessons, practice-problems ### 4. Desmos Graphing Calculator Free browser-based graphing calculator for plotting equations, inequalities, and parametric curves on the coordinate plane. Sliders animate parameters, so you can see how changing slopes, intercepts, or coefficients moves lines, circles, and conics, and check hand-worked analytic geometry problems visually. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.desmos.com/calculator **Tags:** graphing-calculator, coordinate-geometry, function-graphing, conic-sections, visualization ### 5. Math Stack Exchange Real Analysis Tagged question archive where students and mathematicians post proofs and counterexamples on limits, continuity, measure and convergence. Searching it surfaces worked solutions to standard exercises and clarifies where a stalled argument actually breaks down. **Difficulty:** Beginner | **Price:** Free **Link:** https://math.stackexchange.com/questions/tagged/real-analysis **Tags:** real-analysis, proofs, limits, q-and-a ### 6. terrytao.wordpress.com/analysis Terence Tao's Analysis blog is an extensive collection of expository articles, lecture notes, and discussions centered on real analysis and its connections to harmonic analysis, PDE, and additive combinatorics. It features accessible explanations, rigorous proofs, and problem-style posts that illustrate key concepts and techniques in analysis. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://terrytao.wordpress.com/analysis **Tags:** websites, mathematics-statistics, mathematics ### 7. Wolfram MathWorld **Author:** Eric W. Weisstein MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ## Courses ### 1. Real Analysis **Author:** Francis Su Recorded Harvey Mudd College lectures by Francis Su for a first proof-based real analysis course, following Rudin's Principles of Mathematical Analysis: metric spaces, compactness, connectedness, sequences and series, continuity, differentiation and the Riemann-Stieltjes integral. Prepares learners to read and write rigorous analysis proofs. **Difficulty:** Advanced | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PL0E754696F72137EC **Tags:** real-analysis, metric-spaces, proof-based-math, sequences-and-series, riemann-integration ### 2. Real Analysis **Author:** Francis Su Recorded lectures from Francis Su's Math 131 real analysis course at Harvey Mudd College, Spring 2010. Covers the real numbers, metric spaces and topology, sequences and series, continuity, differentiation and integration with full proofs, preparing learners to read and write rigorous analysis arguments. **Difficulty:** Advanced | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PL0E754696F72137EC **Tags:** real-analysis, metric-spaces, sequences-and-series, continuity, proofs ### 3. Real Analysis **Author:** Francis Su Recorded lectures from Francis Su's undergraduate real analysis course at Harvey Mudd College, following Rudin's Principles of Mathematical Analysis: metric spaces, compactness, sequences and series, continuity, differentiation, and the Riemann-Stieltjes integral. Learners gain the proof skills and foundations needed for measure theory. **Difficulty:** Advanced | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PL0E754696F72137EC **Tags:** real-analysis, metric-spaces, riemann-stieltjes-integral, rudin, proofs ### 4. Real Analysis Learn real analysis from Francis Su's engaging course! Explore limits, continuity, and more, building a solid mathematical foundation. **Difficulty:** Advanced | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PL0E754696F72137EC **Tags:** real-analysis, proofs, metric-spaces, lecture-series ### 5. Real Analysis Learn real analysis with Dr. Trefor Bazett! Explore rigorous calculus & mathematical proofs in this comprehensive video course. **Difficulty:** Advanced | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PLBEl4BT8wUgNKTl0bgy6BMQXAShRZor5l **Tags:** real-analysis, sequences-series, epsilon-delta, proofs ### 6. Real Analysis (MIT 18.100A) **Author:** Casey Rodriguez Rigorous foundations of calculus: real numbers, convergence of sequences and series, continuity, differentiability, the Riemann integral, sequences and series of functions, uniform convergence and interchange of limits. Includes 25 lecture videos, notes, problem sets and exams, and teaches how to read and construct proofs. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/ **Tags:** real-analysis, proofs, sequences-and-series, uniform-convergence, riemann-integral ### 7. Multivariable Calculus Master multivariable calculus! Explore functions of several variables, gradients, and more with this comprehensive course. **Difficulty:** Advanced | **Price:** Free **Link:** https://www.khanacademy.org/math/multivariable-calculus **Tags:** calculus, vector-calculus, partial-derivatives, vector-fields, video-lessons ### 8. Real Analysis (MIT 18.100B) **Author:** Tobias Colding Proof-based analysis covering real numbers, limits, convergence, series, continuity, metric spaces, differentiation, Riemann integration and differential equations, with emphasis on proving theorems rigorously. Provides 25 lecture videos (about 33 hours), 26 lecture note files, problem sets and exams. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-100b-real-analysis-spring-2025/ **Tags:** real-analysis, metric-spaces, riemann-integral, mathematical-proofs, sequences-and-series ## Books ### 1. Basic Analysis: Introduction to Real Analysis **Author:** Jiří Lebl A free two-volume undergraduate textbook covering the real numbers, sequences and series, continuity, differentiation, the Riemann integral, sequences of functions, metric spaces and multivariable analysis. Readers learn to read and write the epsilon-delta proofs behind calculus. **Difficulty:** Intermediate | **Price:** Free **Link:** https://www.jirka.org/ra/ **Tags:** real-analysis, riemann-integral, sequences-and-series, metric-spaces, proofs ### 2. Counterexamples in Analysis **Author:** Bernard R. Gelbaum, John M. H. Olmsted Collection of counterexamples in real analysis: pathological functions, sequences, series, sets and integrals that show where standard theorems on continuity, differentiability, integration and convergence fail. Helps the reader test conjectures and understand why each hypothesis in a theorem is needed. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0486428753?tag=edmonddante07-20 **Tags:** real-analysis, counterexamples, continuity, pathological-functions, proofs ### 3. An Introduction to Measure Theory **Author:** Terence Tao Graduate textbook by Terence Tao building Lebesgue measure and integration on the real line and Euclidean space, then abstract measure spaces, product measures and Fubini, and differentiation theorems. Readers learn to prove the core convergence theorems and work rigorously with measurable functions. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://terrytao.files.wordpress.com/2012/12/gsm-126-tao5-measure-book.pdf **Tags:** measure-theory, lebesgue-integration, lebesgue-measure, real-analysis, graduate-mathematics ### 4. Real Analysis - Royden, Fitzpatrick **Author:** H.L. Royden, P.M. Fitzpatrick Graduate text building Lebesgue measure and integration on the real line, then abstract measure spaces, general topology, Banach and Hilbert spaces. Standard preparation for qualifying exams in measure theory and functional analysis. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://amzn.to/3MdQcnd **Tags:** measure-theory, lebesgue-integration, functional-analysis, graduate-mathematics ### 5. Real Mathematical Analysis - Charles Chapman Pugh **Author:** Charles Chapman Pugh An insightful, highly visual, and modern alternative to Rudin for rigorous analysis. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://amzn.to/4rjvvVK **Tags:** real-analysis, metric-spaces, topology, proofs, undergraduate-mathematics ### 6. Principles of Math Analysis - Walter Rudin **Author:** Walter Rudin A notoriously rigorous and concise introduction to real analysis ("Baby Rudin"). **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://amzn.to/3NKoz5E **Tags:** real-analysis, proofs, metric-spaces, riemann-stieltjes-integral ### 7. Understanding Analysis - Stephen Abbott **Author:** Stephen Abbott An exceptionally lucid introduction to real analysis, focusing on the "why" behind calculus. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://amzn.to/49XltTe **Tags:** real-analysis, sequences-series, proofs, undergraduate-mathematics ### 8. Real Analysis: Modern Techniques and Their Applications **Author:** Gerald B. Folland A graduate textbook on measure and integration theory, point-set topology and functional analysis, including Lp spaces, Radon measures, Fourier analysis and distributions. Readers gain the measure-theoretic foundation used in modern analysis and probability. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/1118626397?tag=edmonddante07-20 **Tags:** measure-theory, lebesgue-integration, functional-analysis, lp-spaces, graduate-analysis ### 9. Introduction to Real Analysis **Author:** Robert G. Bartle, Donald R. Sherbert A standard undergraduate textbook building real analysis from the completeness of the reals through sequences, limits, continuity, differentiation, and the Riemann integral. Its graded exercises train students to construct epsilon-delta proofs independently. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0471321486?tag=edmonddante07-20 **Tags:** real-analysis, epsilon-delta-proofs, sequences-and-series, proof-writing --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/real-analysis