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 herds 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



 

                                                            Click here to return to the Frequently Asked Questions page.

                                                             Click here to return to the Home Page.