1. The axioms for the real number system. 2. Countable and uncountable sets. 3. Sequence of real numbers. 4. Nested intervals theorem. 5. Bolzano-Weierstrass theorem, Cauchy criterion and Heine-Borel Theorem. 6. Limits and continuity of functions. 7. Properties of continuous functions.

CILO-1: Explain and use abstract mathematical arguments such as epsilon-delta arguments. CILO-2: Explain the properties of a convergent sequence/series, and use those properties to analyze the convergence of the sequence/series. CILO-3: Explain and apply the important properties of continuous functions in R^1. CILO-4: Explain and apply the elementary theorems in differentiation and integration, and the construction of the Riemann integral.