Nonstandard Real Analysis
Classical or standard analysis is mostly concerned with the study of the real numbers and with the properties of functions defined on them. We shall now describe the use of the hyperreals as valuable tools for mathematical analysis. Through the existence of infinitesimals, finite, and infinite numbers, NSA provides us with a rich structure which we use to formalize alternative treatments of topics in classical analysis. Such treatments are not only valuable for the additional light that they cast on the processes of analysis, but also for the simplification they bring to many concepts and arguments. As will be seen, the mechanization of analysis can benefit directly from this simplification, since difficult instantiation steps in proofs are simply eliminated in many cases. We start by showing how functions defined over the reals and naturals can be systematically extended to the hyperreals and hypernaturals, respectively. These notions are crucial to nonstandard real analysis. We then proceed to develop some elementary analysis that will make use of the new classes of numbers, the infinitely close relation, and other notions induced on them.
KeywordsCauchy Sequence Infinite Series Real Sequence Nonstandard Analysis Transfer Principle
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