TITLE: Inverted induction
QUESTION [13 upvotes]: I am working on a proof, and to do it, I think it would be optimally to use induction backwards.

Show that 1 doesn't work. Assume n doesn't work. Prove that n+1
  doesn't work.

Is this valid?

REPLY [2 votes]: Indeed, it can be viewed as "inverted" induction, i.e. as a special case of Fermat's method of infinite descent, since the contrapositive of your induction step is: $\ n$ works $\,\Rightarrow\, n\!-\!1\,$ works. This descent form of induction is a very natural way to present many inductive proofs.