e are some notes on introductory real analysis. They cover the properties of the real numbers, sequences and series of real numbers, limits of functions, continuity, diferentiability, sequences a d …

Feb 5, 2010 · Section 1.3 is devoted to elementary set the-ory and the topology of the real line, ending with the Heine-Borel and Bolzano-Weierstrass theorems.

A real number x is said to be positive if it has a representative Cauchy sequence (xn)n2N 2 Q+ consisting entirely of positive rational numbers. We denote this by x > 0 or x 2 R +.

Under the identification of Q with a subset of R, the cut defining a real number consists of all rational numbers less than the given real number. The set of cuts gets a natural ordering, given by inclusion.

The foundations of real analysis are given by set theory, and the notion of cardinality in set theory, as well as the axiom of choice, occur frequently in analysis.

In this Chapter 0 we will review in detail the notation and background information that will be assumed throughout Chapters 1–9 of the main text (though we do assume that the reader has a basic …

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