# 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 Archimedeanâ€“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.