I was reading a blogpost here: http://mzargar.wordpress.com/2009/07/19/cauchys-method-of-induction/
One thing that threw me off was that after the first four large displayed equations, there is the statement "it is sufficient to prove that if the theorem holds for $n=m+1$, then it holds for $n=m$."
How is this type of induction valid? I browsed around for things like backward induction, reverse induction, and Cauchy induction, but couldn't find a justification for how this is valid.
With the usual forward induction of verifying a base case and proving $P(n)\implies P(n+1)$, it's easy to intuitively understand how induction will show a property holds for all natural numbers (or at least starting at the base case). But with this reverse induction, it seems to me that if you prove $P(m+1)\implies P(m)$, then if you were able to verify a specific case like $P(15)$, then you would only know it's true for numbers up to $15$. How does it actually prove the property for all naturals?