Difference between revisions of "Cauchy Induction"
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| − | '''Cauchy Induction''' is a beautiful method of Proof by Induction discovered by [[Augustin Louis Cauchy]]. | + | '''Cauchy Induction''' is a beautiful method of Proof by [[Induction]] discovered by [[Augustin Louis Cauchy]]. |
==Definition== | ==Definition== | ||
For a given statement <math>s</math> over the positive integers greater than or equal to 2, the technique of Cauchy Induction is to prove that <math>s(2)</math> is true, and that <math>s(n)</math> implies <math>s(2n)</math>. This implies that <math>S(2^m)</math> is true for all positive <math>m</math>. Then prove that <math>s(n)</math> implies <math>s(n-1)</math>. Then <math>s(n)</math> is true for all <math>n\geq 2</math>. | For a given statement <math>s</math> over the positive integers greater than or equal to 2, the technique of Cauchy Induction is to prove that <math>s(2)</math> is true, and that <math>s(n)</math> implies <math>s(2n)</math>. This implies that <math>S(2^m)</math> is true for all positive <math>m</math>. Then prove that <math>s(n)</math> implies <math>s(n-1)</math>. Then <math>s(n)</math> is true for all <math>n\geq 2</math>. | ||
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Revision as of 13:30, 15 September 2008
Cauchy Induction is a beautiful method of Proof by Induction discovered by Augustin Louis Cauchy.
Definition
For a given statement 
over the positive integers greater than or equal to 2, the technique of Cauchy Induction is to prove that 
is true, and that 
implies 
. This implies that 
is true for all positive 
. Then prove that implies 
. Then is true for all 
.
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