# Concepts of Real Analysis

Schedule of classes: M W F 9:05 AM - 9:55 PM in 173 WILLARD (Sched. \# 597391)

Office Hours: Monday 4:30 - 6 PM & Tuesday 5 - 6:30 PM

FINAL EXAM: MAY 2, 12:20 - 2:10 PM, 101 OSMOND (comprehensive, Exam will cover Chapter 1-4 of the textbook).
REVIEW SESSION WEDNESDAY APRIL 27, FRIDAY APRIL 29 in class. FINAL QUIZ WEDNESDAY APRIL 27.

Finals Week Office Hours: MONDAY 6:00-7:30 PM, TUESDAY 10-11:30 AM.

MIDTERM 2: WEDNESDAY MARCH 16 in class (Sections 2.2 to 2.5, excluding the exponential and areas, perimeters of polygons).
REVIEW SESSION WEDNESDAY MARCH 2 in class.

MIDTERM 1: WEDNESDAY FEB 10 in class (up to Section 2.2.1).
REVIEW SESSION FRIDAY FEB 6 in class.

Syllabus.

Topics: (sections are from the textbook, Brannan's A first Course in Mathematical Analysis)
1. Chapter 1, Real Numbers: eview of properties of real numbers (Sections 1.1.1,1.1.2,1.2.1,1.2.2,1.5), irrational numbers (1.1.3), modulus and distance (1.2.3), triangle inequality, Bernoulli and Cauchy-Schwartz inequalities (1.3), supremum and infimum, completeness (1.4), Archimedean and density properties (1.1.4). SKIP proofs of Least Upper Bound property and density property in 1.4, of existence of roots in 1.5, and proof of Arithmetic-Geometric Mean Inequality in 1.3.
2. Chapter 2, Sequences: monotonic sequences (2.1), null sequences, epsilon-N definition and combination rules (2.2), convergent and divergent sequences, infinite limits, and subsequences (2.3, 2.4), the Monotone Convergence Theorem, Bolzano-Weirerstrass Theorem, e and the exponential function (2.5). SKIP Section 2.5.4, Pi and perimeters/areas of polygons. monotonic sequences (2.1), null sequences (2.2), convergent and divergent sequences (2.3, 2.4), the Monotone Convergence Theorem (2.5).
3. Chapter 3, Series: definition and examples, geometric series, telescoping series, non-null test (3.1), series of positive terms, comparison, root and ratio, Cauchy tests (3.2), alternating series: alternating series test, rearrangements: definition, examples (absolutely and conditionally convergent series), Riemann Rearrangement Theorem (NO proof), rearrangements of absolutely convergence series (NO proof), products: definition, product of absolute convergent series (NO proof) (3.3), the exponential function (SKIP proof the number e is irrational) (3.4).
4. Chapter 4, Continuity: definition, basic examples and properties, discontinuous functions (4.1), properties of continuous functions (4.3), inverse functions of continuous functions (4.3) (we will probably skip 4.4). SKIP Sections 4.2.2 (zeroes of polynomials) and Section 4.4 (more on exponential functions).
5. Chapter 5, Limits
6. definition and examples, connection with continuity of a function (5.1, 5.4.1), epsilon-delta definition (5.3). You will not be tested in the Final Exam on Chapter 5.