Editing Mathematical induction (section)

=== Forward-backward induction ===
{{Main articles|Inequality of arithmetic and geometric means#Proof by Cauchy using forward–backward induction|l1 = Forward-backward induction}}
Sometimes, it is more convenient to deduce backwards, proving the statement for <math>n-1</math>, given its validity for <math>n</math>. However, proving the validity of the statement for no single number suffices to establish the base case; instead, one needs to prove the statement for an infinite subset of the natural numbers. For example, [[Augustin Louis Cauchy]] first used forward (regular) induction to prove the
[[Inequality of arithmetic and geometric means#Proof by Cauchy using forward–backward induction|inequality of arithmetic and geometric means]] for all [[powers of 2]], and then used backwards induction to show it for all natural numbers.<ref>{{Cite web|url=https://brilliant.org/wiki/forward-backwards-induction/|title=Forward-Backward Induction {{!}} Brilliant Math & Science Wiki|website=brilliant.org|language=en-us|access-date=2019-10-23}}</ref><ref>Cauchy, Augustin-Louis (1821). [http://visualiseur.bnf.fr/Visualiseur?Destination=Gallica&O=NUMM-29058 ''Cours d'analyse de l'École Royale Polytechnique, première partie, Analyse algébrique,''] {{Webarchive|url=https://web.archive.org/web/20171014135801/http://visualiseur.bnf.fr/Visualiseur?Destination=Gallica&O=NUMM-29058 |date=14 October 2017 }} Paris. The proof of the inequality of arithmetic and geometric means can be found on pages 457ff.</ref>