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Intro to Analysis in-class_Part_1

# Intro to Analysis in-class_Part_1 - Chapter 0...

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Chapter 0 Pre-preliminaries 0.1 Course Overview The study of the real numbers, R , and functions of a real var, f ( x ) = y where x, y real. Given f : R R which describes some system, how to study f ? Need rigourous vocab for properties of f (definitions) Need to see when some properties imply others (theorems) Result: can make inferences about the system. Motivation for analysis: limits, the heart & soul of calculus. Limits provide a rigourous basis for ideas like sequences, series, continuity, derivatives, integrals. More adv: model an arbitrary function as a limit of a sequence of “nice” functions (polys, trigs) or as a sum of “nice” functions (Fourier, wavelets). All of this requires understanding limits of numbers. Outline: 1. Logic: not, and, or, implication; rules of inference 2. Sets: elements, intersection, union, containment; special sets 3. The real numbers: algebraic properties (+ , × ), order properties ( < ), completeness properties

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8 Math 413 Pre-preliminaries 4. Sequences: types of, convergence, basic results (arithmetic, etc), subsequences, con-
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Intro to Analysis in-class_Part_1 - Chapter 0...

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