Life of Fred: Real Analysis

Real Analysis is a topic studied by math majors in college. This math is more advanced than what most non-majors ever see, but it still has Fred, and he has a very good time with his favorite topic. Real Analysis is the study of real numbers, functions using the real numbers, and the properties of the real numbers and their functions. It is heavy on theorems and proofs.

Major Topics included:
The Real Numbers
Sequences
Series
Tests for Series Convergence
Limits and Continuity
Derivatives
the Riemann Integral
Sequences of Functions
Series of Functions
and Looking Ahead to Topics beyond a First Course in Real Analysis.

Subtopics include:
The axiomatic approach to the real numbers
eleven properties of the real numbers
mathematics after calculus
definition of a function
if a and b are irrational
must ab also be irrational?
two definitions of dense subsets
the natural numbers are well-ordered
the positive real numbers are Archimedean–two definitions
math induction proofs
one-to-one (injective) functions
cardinality of a set
four definitions of onto
finding a one-to-one onto function from (0, 1) to [0, 1]
countable and uncountable sets
Root Test
Ratio Test
Integral Test
absolute and conditional convergence
weak and strong induction proofs
secant lines
limit proofs using e and d
eight theorems about limits and their proofs
lim g(f(x)) does not always equal g(lim f(x))
continuous functions
four theorems about pairs of continuous functions
the squeeze theorem
a very short proof that lim sin x = 0 as x approaches zero
two definitions of derivative
the delta process
the five standard derivative rules and their proofs
how much detail to put in a proof
Schwarzschild radii
converses
contrapositives and inverses
Intermediate Value Theorem
Rolle’s theorem
Mean Value Theorem
L’Hospital’s rule
proving lim (sin ?)/? = 1 in two steps
detailed definition of the Riemann integral
uniform continuity
Fundamental Theorem of Calculus
Cauchy sequence of functions
Cauchy series of functions
uniform convergence of a series of functions
Weierstrass M-test
power series
two formulas for the radius of convergence
taking derivatives and antiderivatives of a power series
Weierstrass Approximation theorem
finding an approximation for ln 5 on a desert island
and the Cantor set.
Calculus is required before studying this topic and Life of Fred: Five Days is highly recommended.

Each of the 113 assignments/puzzles/questions that he gives his students calls for creativity rather than doing drill work. Some of these can be done in a minute. Some will take several hours to complete. They are all meant to be enjoyed. The goal is not to finish the book. It’s just like life.

All answers are included in the textbook.