Instructor: Monica Visan, MS 6167. Email address: email@example.com
TA: Stan Palasek, MS 3965. Email address: firstname.lastname@example.org
Office Hours: Mon: 9:30-10:30am, Wed: 2:30-3:30pm, or by appointment.
Problem session: Tu: 11:00-11:50am in MS 6627.
Textbook: Principles of Mathematical Analysis, 3rd Edition by W. Rudin and Metric Spaces, Cambridge University Press, by E. T. Copson.
Midterm: Wednesday, November 7th, in class.
The midterm counts for 20%
towards the final grade.
Final: Monday, December 10th, 8:00am-11:00am. You have to take the
order to pass the class. The final exam will count for 60% towards the
Homework: There will be weekly homework. It is due on Fridays in class. The homework will count for 20% towards the final grade. Further information is given below.
Grading: Homework 20%; Midterm 20%; Final 60%.
|1||Basic Logic, Peano Axioms, mathematical induction.|
|2||Mathematical induction, equivalence relations, construction of the rational numbers.|
|4||Rational numbers form an ordered field.|
|5||Upper and lower bounds, the least upper bound and the greatest lower bound, the least upper bound property.|
|6||Construction of the real numbers, the Archimedean property and consequences.|
|7||"Uniqueness" of an ordered field with the least upper bound property.|
|8||Sequences: bounded, monotonic, convergent. Examples.|
|9||Limit Theorems. Sequences diverging to ±∞.|
|10||The Cauchy criterion.|
|11||The limsup and liminf of a sequence.|
|12||The limsup and liminf of a sequence.|
|13||Series: convergent, absolutely convergent. The Cauchy criterion, the comparison and root tests.|
|14||The ratio test, the Abel, Leibnitz, and dyadic criterions.|
|16||Functions: domain, range, image, preimage, injective, surjective, bijective, inverse.|
|17||Cardinality of a set: finite, infinite, countable, at most countable, uncountable. Examples.|
|18||A set is infinite if and only if it is equipotent to one of its proper subsets. The Schroder-Bernstein theorem..|
|19||A countable union of countable sets is countable. Finite and countable cartesian products of countable sets. Examples.|
|20||Metric spaces, examples, the Holder and Minkowski inequalities.|
|21||Interior, adherent, isolates, and accumulation points. Open and closed sets and their properties.|
|22||The exterios, boundary, and frontier of a set. Dense and nowhere dense sets. Subspaces of a metric space.|
|23||Complete metric spaces: definition and characterization.|
|24||Complete metric spaces: examples.|
|25||Baire's category theorem.|
|26||Separated sets and properties.|
|27||Connected sets: equivalent definitions.|
|28||Connected subsets of the real line.|
|29||Decomposition of a set into its connected components.|