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Continuity and Topology
Published in James K. Peterson, Basic Analysis II, 2020
We know from the ℜ setting that Cauchy sequences of real numbers must converge to a real number due to the Completeness Axiom. We also know from discussions in (Peterson (21) 2020) that Cauchy sequences in general normed spaces need not converge to an element in the space. Normed spaces in which Cauchy sequences converge to an element of the space are called Complete Spaces. We can easily prove ℜn is a complete space.
Real Numbers
Published in John Srdjan Petrovic, Advanced Calculus, 2020
The Completeness Axiom guarantees the existence of a least upper bound. The mirror image of the set A = (0, 2) in Example 1.2.5 is the interval (−2, 0) and all these negative lower bounds of A become positive upper bounds of (-2, 0). Consequently, the Completeness
Metric Spaces
Published in James K. Peterson, Basic Analysis III, 2020
First, it is time we construct the real numbers. We have always assumed the real numbers satisfy the completeness axiom and that Cauchy sequences of real numbers converge to a real number. Now let’s show this in more detail. Recall
Mathematical Analysis and Optimization for Economists
Published in Technometrics, 2023
Chapters 5 through 7 discuss the global and local extrema of real-valued functions. Global and Local Extrema of Real-Valued Functions gives a general overview of optimization that is a central common concern to economics, business, and management. More often than not, human activities and endeavors are directed toward some maximization or minimization. This is precisely when mathematics becomes both an indispensable tool and a life-saver. Optimization is an everyday seek. Global Extrema of Real-Valued Functions and Local Extrema of Real-Valued Functions focus, respectively, on what they claim. Special emphasis is laid on bounded functions that are ubiquitous in analysis and the first concern regarding the existence of global extrema. It leads to several beautiful destinations such as the completeness axiom, Weierstrauss’ theorem, and the intermediate value theorem. There is a detailed review of the derivative that comes in extremely handy in identifying and classifying points of maxima, points of minima, and points ofinflection.