Real Analysis

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

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

Glossary