Below is a listing of the topics or ideas we'll learn in Real 1. Want to look ahead (or behind)? Look 'em up on Wikipedia or MathWorld.

• The Real Number System: rational, irrational, transcendental numbers, suprema, infima, completeness, nested interval property, denseness of the rational and irrational numbers, cardinality, Cantor-Bernstein-Schroeder Theorem, Cantor diagonalization, Cantor's Theorem.
• Sequences and Series: convergence, algebraic and order limit theorems for sequences and series, monotone convergence theorem, Cauchy condensation test, Bolzano Weierstrass Theorem for squences, Cauchy criterion for convergence of sequences and series, standard tests for convergence of series, rearrangements of series.
• Basic Topology of the Real Line: open sets, closed sets, $G_\delta$ sets, $F_\sigma$ sets, compact sets, Heine-Borel Theorem, perfect sets, connected sets, totally disconnected sets, nowhere dense sets, Cantor's set, Baire Category Theorem.
• Functional Limits and Continuity: the functions of Thomae and Dirichlet, topological and sequential definitions for functional limits, continuity, preservation of connected sets, preservation of compact sets, extreme value theorem, uniform continuity, intermediate value theorem, characterization of set of points of continuity.
• The Derivative: the definition and basic properties, derivatives of the elementary functions, the intermediate value property for derivatives, the mean value theorem, continuity versus differentiability.
• Sequences and Series of Functions: pointwise convergence, uniform convergence and convergence in mean (or probability). uniform convergence preserves continuity, an ocean of examples.
• The Riemann Integral: the geometry of anti-derivatives, the definition of the Riemann integral, properties of the integral, the fundamental theorem of calculus.