What Are Some Innovative Mathematical Strategies Beyond Traditional Induction?

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I was just recently reading about Cauchy's proof of the AM-GM inequality by forward backward induction and was simply blown away by it's elegance. However, I have never seen forward backward induction in any of my discrete maths books (including the one devoted just to mathematical induction). Even the wikipedia page on mathematical induction has no information about backward forward mathematical induction.

I was just wondering if anyone knows of any other innovative strategies like this that are well known. Altough this question is inteneded to find other obscure methods of induction, innovative strategies from other areas of Maths would be appreciated too.
 
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