Life of Fred: Real Analysis
Real analysis is studied after the two years of college calculus. Every math major takes this course.
This book covers all the standard topics in real analysis and adds all the fun that Fred's adventures can bring. Fred pretends he's a WWI bombing pilot (attempting to wipe out the real number line, one point at a time). He shows Kingie how to count ducks' eyes on a lake and then how to herd cattle across the prairie.
All the answers are given in the book.
Chapter 1 The Real Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
the disadvantages of being a biologist or a cook
why we don’t compress real analysis into 50 pages
the axiomatic approach to R
R as unending decimals
eleven properties of the real numbers
mathematics after calculus
open intervals
definition of a function
Nicholas Bourbaki, a famous author who has never been photographed
if a and b are irrational, must a also b 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—one-to-one correspondences
four definitions of onto
closed intervals
finding a one-to-one onto function from (0, 1) to [0, 1] ← not easy!
countable and uncountable sets
Chapter 2 Sequences.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Fibonacci sequence
increasing vs. non-decreasing sequences
bounded sequences
convergent sequences—five definitions
limit of a sequence
tail of a sequence
divergent sequences
maximum member of a set
least upper bound of a set
the Axiom of Completeness for R
the Rabbit and the Wall theorem a.k.a. the Monotone Convergence
theorem (Every bounded non-decreasing sequence is convergent.)
subsequences
a sequence that has subsequences that converge to every natural number
every sequence has a monotone subsequence
Bolzano-Weierstrass theorem
Cauchy sequences
Chapter 3 Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
ø first used for { } in 1939
sigma notation and partial sums
convergence of a series
Cauchy series
arithmetic, geometric, and harmonic series
p-harmonic series
Chapter 4 Tests for Series Convergence. . . . . . . . . . . . . . . . . . . . . . . . . . 83
Tests 1–8 for convergence
non-negative terms and bounded
geometric and |r| < 1
Comparison Test—three forms
alternating series
Root Test
Ratio Test
Integral Test
absolute and conditional convergence
approximating a partial sum
weak and strong induction proofs
when you can rearrange the terms of a series
Chapter 5 Limits and Continuity.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
the idea of limit using a cowboy’s remote control
secant lines
limit proofs—epsilon and delta
eight theorems about limits and their proofs
lim g(f(x)) doesn’t always equal g( lim f(x) )
one formula for the radius of convergence
a second formula for the radius of convergence
interval of convergence
taking derivatives of a power series
taking antiderivatives of a power series
Weierstrass Approximation theorem
finding the coefficients of a power series
finding an approximation for ln 5 on a desert island
continuous functions
four theorems about pairs of continuous functions
composition of functions
the squeeze theorem
Chapter 6 Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
a function as a machine
two definitions of derivative
the delta process
the five standard derivative rules and their proofs
how much detail to put in a proof
breathing as a habit
Schwarzschild radii
converses, contrapositives, and inverses
Intermediate Value Theorem
Rolle’s theorem
Mean Value Theorem
L’Hospital’s rule
proving limit of (sin x)/x = 1 as x → 0 in two steps ← amazing
Chapter 7 The Riemann Integral.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
the four stages of learning about integrals in calculus
detailed definition of the Riemann integral (many pages)
uniform continuity
Fundamental Theorem of Calculus
Chapter 8 Sequences of Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
if each fn is continuous and fn → f, will f be continuous?
if each fn is differentiable and fn → f, will f be differentiable?
if each fn is continuous and fn → f uniformly, will f be continuous?
if each fn is differentiable and fn → f uniformly, will f be differentiable?
Cauchy sequence of functions
Chapter 9 Series of Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
how sequences of numbers, series of numbers, and sequences of
functions all make series of functions an easier topic
Cauchy series of functions
uniform convergence of a series of functions
Weierstrass M-test
power series
Chapter 10 Looking Ahead. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
working in 4th dimensional real numbers
Cantor set
definition of dimension
The 0.63092975357145743709952711434276th dimension
Riemann–Stieltjes integrals
Lebesgue integrals
measure of a set
metric spaces
foundations of set theory
topology
abstract arithmetic—the axioms and derivation of N , Z, Q, and R
modern (abstract) algebra—semigroups, monoids, groups, rings, fields
linear algebra
Solutions to All the Puzzles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
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