1.    Real number system (1.5 weeks): algebraic properties, completeness axiom, least upper bound property, Archimedean property, cardinality of sets, countability of rational numbers.

2.    Real sequences and series (2.5 weeks): limits, Cauchy sequences, upper and lower limits, squeeze theorem, tests for convergence of sequences and series, Abel summation formula, Dirichlet test.

3.    Limit and continuity of functions (3 weeks): the e-d definition for limits and continuity, the intermediate value theorem and extreme value theorem for continuous functions over finite closed intervals, uniform continuity.

4.    Differentiation on ¡ (2.5 weeks): Derivatives and their basic rules, higher derivatives, the mean value theorem, Taylor’s theorem, L’Hospital’s rule, inverse function theorem (for one variable), Newton-Raphson’s method.

5.    Integration on ¡ (2.5 weeks): the theory of Darboux-Riemann integrals over ¡, change of variables, the fundamental theorem of calculus, integration by parts, improper integrals.

6.    Function sequences and series (3 weeks): Pointwise convergence vs. uniform convergence, Weierstrass M-test, the connection between uniform continuity and continuity, differentiability, and integrability, Weierstrass approximation theorem, real analytic functions.

For a 16-week semester, 0.5 weeks are available to each of the midterm and the final exam.

1.    Wade, An Introduction to Analysis, 4th edition.

2.    Rudin, Principles of Mathematical Analysis, 3rd edition.

3.    Apostol, Mathematical Analysis, 2nd edition.

4.    Pugh, Real Mathematical Analysis, 2nd edition.

5.    Tao, Analysis I, 4th edition.