Techniques of Elementary Analysis
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We begin by using a form of interval arithmetic as a foundation for the construction of the real number line ℝ. Having discussed the elementary algebraic and order-theoretic properties of real numbers, we prove that ℝ is complete in two senses: Dedekind (order) complete and Cauchy (sequentially) complete. The next step is to generalise from the reals to metric and normed spaces. A particularly important property for us is total boundedness, which plays a big part in proving the existence of suprema and infima; for that reason we need to prove that there are lots of totally bounded subsets in a totally bounded space. We also highlight the important property of locatedness for subsets of a metric space, a property that holds automatically in classical mathematics.
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