· 6 min
The geometry that makes minimisation tractable. Convex sets and functions and their first and second-order characterisations, the separating and supporting hyperplane theorems from projection onto a closed convex set, and the fact that a convex function's local minimum is global.
· 7 min
Infinite sums and the functions they define. Convergence of series, absolute convergence and the root test, the radius of convergence of a power series by Cauchy-Hadamard, term-by-term differentiation, and the exponential series with its defining identity.
· 6 min
Area as a limit of sums squeezed between over and under estimates. Upper and lower sums, the Riemann criterion for integrability, the integrability of continuous functions through uniform continuity, and the fundamental theorem of calculus tying the integral to the derivative.
· 5 min
The derivative and what the mean value theorem extracts from it. The derivative as a limit, differentiability implying continuity, Fermat's interior-extremum principle, Rolle's theorem and the mean value theorem, and Taylor's theorem with the Lagrange remainder.
· 4 min
Absolute continuity, equivalence, and the density that connects two measures.
· 7 min
The Lebesgue integral built from simple functions, and the three convergence theorems that make it usable. We prove monotone convergence, deduce Fatou and dominated convergence, and record the L^p inequalities.
· 8 min
How size is assigned to sets. Sigma-algebras and measures, continuity along monotone limits, closure of measurable functions under pointwise limits, the Caratheodory extension theorem that turns an outer measure into a measure, and the construction of Lebesgue measure.
· 8 min
Distance, abstracted. Metric and normed spaces, open and closed sets, completeness and the Banach fixed-point theorem, compactness and Heine-Borel, the equivalence of all norms in finite dimensions, and that a continuous function on a compact set attains its extremes.
· 6 min
What it means for a function to have no jumps. Limits of functions and their sequential characterisation, continuity, the intermediate value theorem from completeness, the extreme value theorem from Bolzano-Weierstrass, and uniform continuity on a closed interval.
· 8 min
The completeness of the real numbers and what it proves. The least upper bound axiom, limits of sequences, the monotone convergence theorem, Cauchy sequences, and the Bolzano-Weierstrass theorem, the four equivalent faces of completeness that the rest of analysis rests on.