Real Analysis I

Catalog description

Calculus is developed rigorously from the ground up, beginning with ordered field axioms. Basic algebraic properties are derived followed by elementary results for subsets of R, including inf, sup, and transfinite sets. On this foundation, we define sequences and their limits, covering the Bolzano-Weierstrass theorem, Cauchy sequences, liminf, limsup. Similar results for limits of real-valued functions then lead to the derivative and the Riemann integral. For derivatives, we cover the mean value theorem, I'Hopital's Rul, and inverse functions, and for the integral, we cover Riemann sums, the mean value theorem for integrals, and the fundamental theorem of calculus