## Math 413: Introduction to Analysis

Mon, Wed, Fri 2:00 - 2:50pm -- Thackeray 704

Homework
• Instructor. Dr. Marta Lewicka (office hours in Thackeray 408, Monday 3pm - 4pm)
Teaching Assistant. Marc Beauchamp. Recitations in Thackeray 704 from 1:00pm to 1:50pm on Tue and Th.
• Textbook. Jiri Lebl: Basic Analysis. Custom Pitt edition.

• Prerequisites. The course covers the foundations of theoretical mathematics and analysis. The principal topics of the course include fundamentals of logic, sets, functions, number systems, order completeness of the real numbers and its consequences, and convergence of sequences and series of real numbers. Successful completion of Math 0230 (Calculus II) or equivalent is required to follow this course. If you do not feel comfortable with the prerequisite material, please contact the instructor in the beginning of the course.

• Homework. Homework will be assigned each Thursday (starting in the second week of the semester), and it will be due the following Thursday at the recitations. Late homework will not be accepted. The solution of each exercise will be evaluated in the scale 0-5 points, taking into account the correctness, clarity and neatness of presentation. You should solve the problems and write up solutions independently.

• Core topics.
1. Logic, proofs and quantifiers. Basic set theory. Functions. Equivalence relations.
2. Elementary properties of the natural numbers; mathematical induction.
3. Axiomatic introduction to the ordered fields of rational and real numbers.
4. Elementary inequalities.
5. The Completeness Axiom; Archimedean Property of the real numbers; density of the rational and irrational numbers in the real numbers.
6. Countability of the rationals; decimal expansions of real numbers; uncountability of the real numbers.
7. Sequences and an introduction to series; the geometric series; limits; Limit Laws.
8. The Monotone Convergence Theorem.
9. The Bolzano-Weierstrass Theorem.
10. Cauchy sequences; Cauchy completeness of the real numbers.
11. Series; convergence tests; alternating series; conditional convergence and rearrangements.
12. Cluster points; limits of functions; continuous functions and examples of sets of points of discontinuity.

• Calendar.  4 Jan (Wed): First class 16 Jan (Mon): No class (Martin Luther King's day) 6, 8, 10 Mar (Mon, Wed, Fri): Spring break - no classes 21 Apr (Fri): Last class 29 Apr (Saturday) 8:00AM - 9:50AM: Final Exam (in Thack 704)