Life of Fred Mathematics

Life of Fred: Real Analysis
(For Math Majors)

Life of Fred:
Real Analysis


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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.

• 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.

• 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.

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Life of Fred Real Analysis is a hardcover textbook containing 302 pages. This book is not consumable. All answers are written on separate paper or in a notebook.

Number of Lessons: 113 assignments over 10 Chapters. It is not broken up into daily lessons.