structural:reformatting
Forward-backward induction

Sometimes, it is more convenient to deduce backwards, proving the statement for n − 1 {\displaystyle n-1}, given its validity for n {\displaystyle n}. 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 for all powers of 2, and then used backwards induction to show it for all natural numbers.