= Solution
<Mathematical induction> has two obligations. Establish the base case $S(0)$ if the <natural numbers> start at zero, or $S(1)$ if they start at one. Then prove the induction step: for an arbitrary natural $k$ in the relevant range, assume $S(k)$ and deduce $S(k+1)$.
The base case starts the chain, and the step propagates truth to every subsequent <natural number>:
$$
\boxed{S(n_0)\ \land\ \bigl(\forall k\geq n_0,\ S(k)\Rightarrow S(k+1)\bigr)\ \Longrightarrow\ \forall n\geq n_0,\ S(n).}
$$
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