# 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.

Approved: 2013-07-01, John Hunter
Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Stephen Abbott, Understanding Analysis, \$53; Alternate text: Kenneth Ross, Elementary Analysis, ISBN 146146703, \$50
Search by ISBN on Amazon: 1441928669
Prerequisites:
MAT 21B
Course Description:
Students develop skills required to understand rigorous definitions in analysis and to construct and write proofs. They learn the axiomatic definition of the real numbers as a complete ordered field and the basic topology of the real numbers. They study the convergence of sequences and series.
Suggested Schedule:
 Lecture(s) Sections Comments/Topics 3 1.1, 1.2 Sets and functions. Logic and proofs. Proof by induction. 1 1.2, 8.4 Algebraic and order axioms for the real numbers. Absolute values. 1 1.3 Completeness axiom for the real numbers. Suprema and infima. 2 1.4 Archimedian property of the real numbers. Density of the rational numbers. 3 2.1, 2.2 Sequences. Definition of the limit of a sequence. 3 2.3 Algebraic and order limit theorems. 2 2.4 Monotone convergence. The limsup and liminf. 2 2.5 Subsequences. Bolzano-Weierstrass theorem. 1 2.6 Cauchy sequences. 3 2.7 Infinite series. Absolute convergence. Comparison test. Alternating series test. 1 2.8 Double summations. Products of infinite series. 3 3.1, 3.2 Topology of the real numbers. Open and closed sets. Accumulation, boundary, and interior points. 2 3.3 Compact sets of real numbers. Heine-Borel theorem. Finite intersection property. 1 3.4 Connected and disconnected sets of real numbers.