Subsection A.1 Strong Induction
We have an equivalent statement of the Principle of Mathematical Induction that is often very useful.
Principle A.1.4. Second Principle of Mathematical Induction.
Let \(S(n)\) be a statement about integers for \(n \in {\mathbb N}\) and suppose \(S(n_0)\) is true for some integer \(n_0\text{.}\) If \(S(n_0), S(n_0 + 1), \ldots, S(k)\) imply that \(S(k + 1)\) for \(k \geq n_0\text{,}\) then the statement \(S(n)\) is true for all integers \(n \geq n_0\text{.}\)