# Life of Fred: Real Analysis

**Description**

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.