# Using Continuity Induction

by Dan Hathaway

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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