Outline

The Real Analysis Proficiency Exam covers material from a standard two-semester undergraduate course, including the following:

1.

Properties of the real numbers

(a)

Real numbers, rational and irrational numbers

(b)

The denseness of the rationals

(c)

The completeness axiom (``least upper bound'' property)

2.

Sequences

(a)

Definition of limit(finite and infinite), basic limit theorems

(b)

Monotone sequences

(c)

Bolzano-Weierstrass theorem

(d)

Cauchy sequences and completeness

(e)

Subsequences, limsup and liminf

(f)

Series and partial sums, including convergence tests

3.

Functions

(a)

Continuity and uniform continuity

(b)

Pointwise and uniform convergence

(c)

Power series and radius of convergence

4.

Differentiation

(a)

Basic rules (chain rule, product rule)

(b)

Mean Value Theorem, Taylor's Theorem

(c)

Taylor series and power series

5.

Integration

(a)

Riemann integral

(b)

Basic theorems (continuous functions are integrable, Integral Mean Value Theorem, etc.)

(c)

Fundamental Theorem of Calculus

(d)

Improper integrals

These topics are covered in many standard undergraduate real analysis textbooks, for example:

• Kenneth A. Ross, Elementary Analysis: The Theory of Calculus, Springer.
• Jerrold Marsden and Michael Hoffman, Elementary Classical Analysis, Freeman Pub. Co.
• William Wade, An Introduction to Analysis, Prentice Hall.

In particular, the student should know the statement and proof of the following theorems and results:

1.

Bounded monotone sequences converge.

2.

Convergent sequences are Cauchy sequences.

3.

The Bolzano-Weierstrass theorem

4.

Absolutely convergent series converge.

5.

Continuous functions on a closed interval attain their maximum.

6.

The intermediate value theorem

7.

The uniform limit of continuous functions is continuous.

8.

Rolle's theorem or the Mean Value Theorem

9.

If a function is monotone on [a,b], then it is Riemann integrable on [a,b].

10.

The fundamental theorem of integral calculus