Real Analysis I
Revised: September, 2014
Sequences of real numbers, continuous functions, and differentiation. Prerequisite:
MATH 250 and MATH 255. Three Semester Hours.
Math250 and Math255.
Student Learning Objectives
By the end of the course students will be able to:
- Use the definitions of convergence as they apply to sequences, series, and functions;
- Determine the continuity, differentiability, and integrability of functions defined
on subsets of the real line;
- Produce rigorous proofs of results that arise in the context of real analysis; and
- Write proofs of theorems that meet rigorous standards based on content, precision,
Real Analysis, A First Course, Second Edition. Addison-Wesley, 2002.
Grading procedures and factors influencing course grade are left to the discretion
of individual instructors, subject to general university policy.
Attendance policy is left to the discretion of individual instructors, subject to
general university policy.
Chapter 1: Real Numbers (6 days)
Completeness; countable and uncountable sets; real valued functions
Chapter 2: Sequences (10 days)
Convergent monotone and Cauchy sequences; subsequences; Bolzano-Weierstrass
Chapter 3: Limits and Continuity (14 days)
Limit theorems; one-sided and infinite limits; continuous functions; intermediate
and extreme values; uniform continuity; monotone functions
Chapter 4: Differentiation (10 days)
The definition and rules of differentiation; mean value and L'Hopital
Chapter 5: Integration (5 days, if time allows)
Riemann Integral; conditions for Riemann integrability
Times above include review and testing.