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 (email@example.com)
to set up an appointment or just drop by (Wean Hall 8206).
GRADER: Yuanyuan Feng (firstname.lastname@example.org) in Wean Hall 7205.
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.