## Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 127A: Real Analysis

**Approved:**2015-11-06, John Hunter and Janko Gravner

**Suggested Textbook:**(actual textbook varies by instructor; check your instructor)

Stephen Abbott, Understanding Analysis, 2nd ed., Springer 2015. (Alternative: Kenneth Ross, Elementary Analysis, Springer, 2nd ed., Springer, 2013.)

https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/intro_analysis.html

https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/intro_analysis.html

**Course Description:**

Real numbers, sequences, series, and continuous functions.

**Suggested Schedule:**

Suggested number of lectures are indicated in parentheses.

- (3) Axioms for the real numbers. Supremum and infimum. Archimedean property. Density of rationals.
- (3) Sequences. Limits. Algebraic properties. Definition of subsequences.
- (3) Convergence of monotone sequences. Extended real line. Limsup and liminf.
- (1) Cauchy sequences.
- (1) Bolzano-Weierstrass theorem.
- (4) Series. Convergence and absolute convergence. Comparison test. Ratio and root tests. Alternating series. Rearrangements.
- (3) Topology of real numbers. Open, closed, and connected sets. Accumulation, boundary, and interior points.
- (2) Compact sets of real numbers. Heine-Borel theorem.
- (1) Limits of real functions.
- (1) Continuous functions. Algebraic properties.
- (1) Definition of uniform continuity.
- (2) Continuous functions on compact sets. Extreme value theorem. Uniform continuity.
- (1) Intermediate value theorem.

Total: 26 lectures