The backwards mathematical induction is Let $n$ be an natural number and $P(m)$ be a property, and if $P(m+1)$ is true, then $P(m)$ is true. The problem is given that $P(n)$ is true, prove $\forall m\in\mathbb{N}$ and $m\leq n$, $P(m)$ is true.
What make me puzzled is the hint in the book as I should make induction according to $n$, but I think that the $n$ is fixed? What’s the meaning of changing $n$?
You could find the problem on tao’s analysis.