# 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
Prerequisites:
(MAT 021C or MAT 021CH); (MAT 067 or (MAT 022A or MAT 027A or BIS 027A), MAT 108).
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