Life of Fred: Real Analysis

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,
• Schwarzschild radii,
• 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.

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