R E A L    A N A L Y S I S   I  -  F A L L   2 0 1 4 

LECTURES: Monday, Wednesday, Friday 2:30pm-3:20pm, in Wean Hall 4623

INSTRUCTOR: Solesne Bourguin, office hours Monday, Friday 4:00pm-5:00pm, you also can email me (bourguin@math.cmu.edu) to set up an appointment or just drop by (Wean Hall 8206).

GRADER: Yuanyuan Feng (yuanyuaf@andrew.cmu.edu) in Wean Hall 7205.

COURSE DESCRIPTION: It is intended to expand on topics introduced in the calculus sequence and to consider them at a higher mathematical level. The material covered will include: The natural number system, the integer and rational number systems: construction, definitions and main properties. The real number system: Field and order axioms, sups and infs, completeness, integers and rational numbers. Real sequences: Limits, accumulation points, limsup and liminf, subsequences, monotonic sequences, Cauchy's criterion, the Bolzano-Weierstrass theorem. Topology of the real line: Open sets, closed sets, density, compactness, the Heine-Borel theorem. Continuity: Attainment of extrema, the intermediate value theorem, uniform continuity. Differentiation: Chain rule, local extrema, mean-value theorems, L'Hospital's rule, Taylor's theorem. Riemann integration: Partitions, upper and lower integrals, sufficient conditions for integrability, the fundamental theorem of calculus. Sequences of Functions: Pointwise convergence, uniform convergence, interchanging the order of limits.

PREREQUISITES: 21-122 and 21-127.

TEXTBOOK: Our reference text will be Real Analysis and Foundations, Third Edition, by Krantz.

HOMEWORK: Every Wednesday for the next Wednesday.

GRADING: problem sets (25%), midterm (35%) and a final exam (40%).

Construction of the reals as a quotient ring.
Precisions on the Riemann-Weierstrass Rearrangment Theorem.

Problem set 1.
Problem set 2.
Problem set 3.
Problem set 4.
Problem set 5.
Problem set 6.
Problem set 7.
Problem set 8.
Problem set 9.

Review Exercises.