Course Description:
This course is an introduction to writing mathematical proofs, including discussion
of mathematical notation, methods of proof, and strategies for formulating
and communicating mathematical arguments.
Class Information:
3 one hour lectures per week
2 one hour recitations per week
3 Exams
Weekly written homework assignments
Weekly in-class presentations
Avg. Class Size: 25 students
Textbook:
Personal Notes.
Basic Analysis by Jiří Lebl. (Loosely used)
Topics Covered:
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Chapter 1: Logic, proofs and quantifiers. Basic set theory. Functions. Equivalence relations.
Elementary properties of the natural numbers; mathematical induction.
Axiomatic introduction to the ordered fields of rational and real numbers.
Elementary inequalities.
The Completeness Axiom; Archimedean Property of the real numbers; density
of the rational and irrational numbers in the real numbers.
The Completeness Axiom; Archimedean Property of the real numbers; density
of the rational and irrational numbers in the real numbers.
Countability of the rationals; decimal expansions of real numbers; uncountability
of the real numbers.
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Chapter 2: Sequences and an introduction to series; the geometric series; limits; Limit Laws.
The Monotone Convergence Theorem. The Bolzano-Weierstrass Theorem. Cauchy sequences;
Cauchy completeness of the real numbers. Series; convergence tests; alternating series;
conditional convergence and rearrangements.
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Chapter 3: Cluster points; limits of functions; continuous functions and
examples of sets of points of discontinuity.