# Life of Fred: Real Analysis

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.

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## Description

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 ε and δ,
• 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,
• converses, contrapositives, and inverses,
• Intermediate Value Theorem,
• Rolle’s theorem,
• Mean Value Theorem,
• L’Hospital’s rule,
• proving lim (sin θ)/θ = 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.