Real Analysis

Interactive visualizations for learning the ideas of real analysis by moving them yourself.

• 1. Foundations of Proof — The language of analysis, from the ground up — propositions and quantifiers through direct proof, contrapositive, contradiction, strong induction, equivalence relations, and function proofs, following the arc of Hammack’s Book of Proof.
• 2. Number Systems — From naturals to reals — building the number systems from axioms, and telling countable infinities from uncountable ones.
• 3. Sequences — Convergence of sequences via ε-N arguments: monotone and Cauchy sequences, limsup and liminf.
• 4. Continuity — Continuity via ε-δ, uniform continuity, and the great theorems: the intermediate and extreme value theorems.
• 5. Topology — Open sets, compactness, completeness — the topology of the real line, the Baire category theorem, Cantor sets.
• 6. Approximation — Approximating continuous functions by polynomials and taming families of functions — the core approximation theorems.
• 7. Series — Watch infinite series converge or diverge through their partial sums, and master the convergence tests.
• 8. Power Series — Inside the radius of convergence — Abel and Tauber theorems, rearrangements, Cauchy products.
• 9. Series of Functions & Fourier — When series of functions converge uniformly — plus Fourier series and the Gibbs phenomenon.
• 10. Differentiation — From the definition of the derivative through the mean value theorem, l'Hôpital, and Taylor expansions.
• 11. Integration — From the Riemann integral to the Lebesgue integral — how the idea of integration expands.

Glossary