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swanson i introduction analysis complembers 2020

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Textbook in PDF format These notes were written expressly for Mathematics 112 at Reed College, with first use in the spring of 2013. The title of the course is “Introduction to Analysis”. The prerequisite is calculus. I maintain two versions of these notes, one in which the natural, rational and real numbers are constructed and the Least upper bound theorem is proved for the ordered field of real numbers, and one version in which the Least upper bound property is assumed for the ordered field of real numbers. You are reading the longer, former version. Preface The briefest overview, motivation, notation How we will do mathematics Statements and proof methods Statements with quantifiers More proof methods Logical negation Summation Proofs by (mathematical) induction Pascal's triangle Concepts with which we will do mathematics Sets Cartesian product Relations, equivalence relations Functions Binary operations Fields Order on sets, ordered fields What are the integers and the rational numbers? Increasing and decreasing functions Absolute values Construction of the number systems Inductive sets, a construction of natural numbers Arithmetic on ℕ₀ Order on ℕ₀ Cancellation in ℕ₀ Construction of ℤ, arithmetic, and order on ℤ Construction of the ordered field ℚ Construction of the field ℝ of real numbers Order on ℝ, the Least upper bound theorem Complex numbers Functions related to complex numbers Absolute value in ℂ Polar coordinates Topology on the fields of real and complex numbers The Heine–Borel theorem Limits of functions Limit of a function When a number is not a limit More on the definition of a limit Limit theorems Infinite limits (for real-valued functions) Limits at infinity Continuity Continuous functions Topology and the Extreme value theorem Intermediate value theorem Radical functions Uniform continuity Differentiation Definition of derivatives Basic properties of derivatives The Mean value theorem L'Hôpital's rule Higher-order derivatives, Taylor polynomials Integration Approximating areas Computing integrals from upper and lower sums What functions are integrable? The Fundamental theorem of calculus Integration of complex-valued functions Natural logarithm and the exponential functions Applications of integration Sequences Introduction to sequences Convergence of infinite sequences Divergence of infnite sequences and infinite limits Convergence theorems via epsilon-N proofs Convergence theorems via functions Bounded sequences, monotone sequences, ratio test Cauchy sequences, completeness of ℝ, ℂ Subsequences Liminf, limsup for real-valued sequences Infinite series and power series Infinite series Convergence and divergence theorems for series Power series Differentiation of power series Numerical evaluations of some series Some technical aspects of power series Taylor series Exponential and trigonometric functions The exponential function The exponential function, continued Trigonometry Examples of L'Hôpital's rule Trigonometry for the computation of some series Appendix A: Advice on writing mathematics Appendix B: What one should never forget Index

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