> [!NOTE] Proposition (Induction Principle, Strong Form) > Suppose that $(P(n))_{n\geq n_{0}}$ is a sequences of properties, one for each natural number, such that > 1. if $P(n_{0}),P(n_{0}+1),\dots,P(n_{0}+m)$ are true for some positive integer $n_{0}$ and non-negative integer $m,$ and > 2. if for every $k>n_{0}+m,$ $P(j)$ true for all $n_{0}\leq j<k$ implies $P(k)$ true, > >then $P(n)$ is true for all $n\geq n_{0}.$