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Using Continuity Induction

by Dan Hathaway

This article originally appeared in:
College Mathematics Journal
May, 2011

Subject classification(s): Analysis
Applicable Course(s): 4.11 Advanced Calc I, II, & Real Analysis

Here is a technique for proving the fundamental theorems of analysis that provides a unified way to pass from local properties to global properties on the real line, just as ordinary induction passes from local implication (if true for \(k\), the theorem is true for \(k + 1\)) to a global conclusion in the natural numbers. The author demonstrates this method by proving the Intermediate Value Theorem and the Heine-Borel Theorem.

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