Instructor: Alena Erchenko

Course on canvas.dartmouth.edu.

## Syllabus

 Date Topic References 01/03 Introduction. Basic Set Theory. (R) Chapter I(L) Introduction, Section 0.3 01/05 Bijections, invertible maps. Cardinality. Cantor's theorem. (L) Introduction, Section 0.3 01/08 Ordered fields (R) Chapter II, Section 1-2(L) Section 1.1 01/10 Least upper bound. Existence of square roots (R) Chapter II, Section 3-4(L) Section 1.2 01/11 (block 10X, 12:15pm - 1:05 pm ET) Metric spaces (R) Chapter III, Sections 1-2(L) Section 7.1 01/12 Ball neighborhoods. Open sets (R) Chapter III, Section 2 (L) Section 7.2 01/15 No class -> Moved to X-hour on 01/11 01/17 Open and closed sets. (R) Chapter III, Sections 2 (L) Section 7.2 01/19 Convergence. (R) Chapter III, Section 3 (L) Section 7.3 01/22 Complete spaces. (R) Chapter III, Section 4 (L) Section 7.4 01/24 Compact sets (R) Chapter III, Section 5 (L) Section 7.4.2 01/26 Equivalent definitions of compactness. (R) Chapter III, Section 5 (L) Section 7.4.2 01/29 Heine-Borel. Connected sets (R) Chapter III, Section 6(L) Sections 7.4.2 and 7.2.2 01/31 Continuous function. Functions on connected spaces. (R) Chapter IV, Sections 1, 5 02/02 Intermediate Value Theorem. Limits (R) Chapter IV, Section 2, 5 02/05 Rational functions (R) Chapter IV, Section 3 02/07 Functions on a compact (R) Chapter IV, Section 4 02/09 Sequences of functions (R) Chapter IV, Section 6 02/12 Derivatives (R) Chapter V, Section 1-2 02/14 Mean Value Theorem (R) Chapter V, Section 3 02/15(block 10X, 12:15pm - 1:05 pm ET) Taylor's Theorem (R) Chapter V, Section 4 02/16 Riemann Integral (R) Chapter VI, Section 1 02/19 Properties of integral (R) Chapter VI, Sections 2-3 02/21 Integrable functions 02/23 Fundamental Theorem (R) Chapter VI, Sections 4-5 02/26 Interchange of limit operations (R) Chapter VII, Section 1 02/28 Power series (R) Chapter VII, Sections 2-3 03/01 The fixed point theorem (R) Chapter VIII, Section 1 03/04 Review 03/08(11:30am-2:30pm ET) Final exam