Elements of real analysis
Focusing on one of the main pillars of mathematics, Elements of Real Analysis provides a solid foundation in analysis, stressing the importance of two elements. The first building block comprises analytical skills and structures needed for handling the basic notions of limits and continuity in a simple concrete setting while the second component involves conducting analysis in higher dimensions and more abstract spaces. Largely self-contained, the book begins with the fundamental axioms of the real number system and gradually develops the core of real analysis. The first few chapters present the essentials needed for analysis, including the concepts of sets, relations, and functions. The following chapters cover the theory of calculus on the real line, exploring limits, convergence tests, several functions such as monotonic and continuous, power series, and theorems like mean value, Taylor's, and Darboux's. The final chapters focus on more advanced theory, in particular, the Lebesgue theory of measure and integration. Requiring only basic knowledge of elementary calculus, this textbook presents the necessary material for a first course in real analysis. Developed by experts who teach such courses, it is ideal for undergraduate students in mathematics and related disciplines, such as engineering, statistics, computer science, and physics, to understand the foundations of real analysis.
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absolutely convergent algebra assume axioms bijection Borel set Cauchy criterion Cauchy sequence Chapter choose clearly closed cluster point conclude continuous function converges uniformly Corollary countable cr-algebra Definition 5.1 denoted derivative differentiable disjoint divergent elements equation evaluate Exercise exists f(xn Figure finite number fn(x function defined function f G a,b Give an example given hence implies improper integral induction inequality infinite injective Lebesgue integral Lebesgue measurable Lemma Let xn lim f(x lim inf lim sup limit limxn mean value theorem monotonic natural number neighborhood Note obtain open interval open set outer measure partition pointwise polynomial positive integer positive number Proof properties rational number real numbers result Riemann integrable satisfies sequence xn sinx strictly increasing subinterval subsequence of xn subset Suppose uniform convergence uniformly continuous upper bound xi+i xn+i xn+l