Syllabus

Week 1: August 22 - 26.

II.1 The field properties.

II.2 Order.

II.3 The least upper bound property.

II.4 The existence of square roots.

Week 2: August 29 - September 2.

III.1 Definition of metric space. Examples.

III.2 Open and closed sets.

Characterization of open sets in R.

Week 3: September 5 - 9.

III.3 Convergent sequences.

Convergent sequences in R.

Week 4: September 12 - 16.

Sequences of extended real numbers.

Monotonic sequences; limits infimum and supremum.

Week 5: September 19 - 23.

III.4 Completeness.

The Bolzano-Weierstrass Theorem.

Countable Sets.

The Baire Category Theorem.

Week 6: September 26 - 30.

Separability.

The Lindelöf Theorem.

Exam 1: Thursday, September 29.

Week 7: October 3 - 7.

III.5 Compactness.

The Heine-Borel Theorem.

III.6 Connectedness.

Week 8: October 10 - 14.

IV.1 Definition of continuity. Examples.

IV.2 Continuity and limits.

IV.3 The continuity of rational operations.

Week 9: October 17 - 21.

IV.4 Continuous functions on a compact metric space.

Uniform continuity.

IV.5 Continuous functions on a connected metric space.

IV.6 Sequences of functions.

Uniform Convergence.

Week 10: October 24 - 28.

The Arzela-Ascoli Theorem.

V.1 The definition of derivative.

V.2 Rules of differentiation.

Course Withdrawal Deadline: Friday, October 28.

Week 11: October 31 - November 4.

V.3 The Mean Value Theorem.

V.4 Taylor's Theorem.

VI.1 Riemann integration: definitions and examples.

VI.2 Linearity and order properties of the integral.

Week 12: November 7 - 11.

VI.3 Existence of the integral.

Exam 2: Thursday, November 10.

Week 13: November 14 - 18.

VI.4 The Fundamental Theorem of Calculus.

VI.5 The logarithmic and exponential functions.

The Weierstrass Approximation Theorem.

Week 14: November 21 - 23.

VII.1 Integration and differentiation of sequences of functions.

Thanksgiving Break: November 24 - 25.

Week 15: November 28 - December 2.

VII.2 Infinite series.

VII.3 Power series.

Week 16: December 5 - 6.

Taylor series.