Real Analysis

This course contains 40 visible knowledge chapters in nine content-sized units of lengths 4, 6, 4, 4, 4, 4, 4, 6, and 4. Why do the rules of calculus actually work, and exactly when do they fail? Start with the real number line, upper bounds, nested intervals, and ordinary sequences. The course reads ε, δ, N, limsup, closure, and every new symbol aloud before using them, then turns “gets close,” “stays bounded,” “has no gaps,” and “one error bound works everywhere” into statements that can be proved. It now proves the complete sequence bridge explicitly: bounded data yields a convergent subsequence by Bolzano–Weierstrass, and Cauchy control plus that subsequential limit forces the full real sequence to converge. From those foundations it rebuilds series with tail certificates, compactness, continuity, derivatives including Darboux’s theorem, Riemann integration beyond continuous functions, uniform function series, completeness of continuous-function space, Arzelà–Ascoli subsequence extraction, and combined error budgets. Every theorem shows which assumption does the work, and the final study proves a root exists and is unique, bounds a bisection error, and justifies an exact integral one dependency at a time.

Before this course: Completed Calculus and Proof, Logic & Set Theory. The course assumes limits, derivatives, Riemann-sum intuition, elementary series, quantified proof, sets, functions, countability, induction, and contradiction, but no topology, measure theory, complex analysis, or abstract algebra.