\section{Mathematical Induction}

\begin{theorem}[First Principle of Mathematical Induction]
	Let $S$ be a set of integers containing $a$. Suppose $S$ has the property that whenever some integer $n \geq a$ belongs to $S$, then the integer $n + 1$ also belongs to $S$. Then, $S$ contains every integer greater than or equal to $a$.
\end{theorem}

\begin{theorem}[Second Principle of Mathematical Induction]
	Let $S$ be a set of integers containing $a$. Suppose $S$ has the property that $n$ belongs to $S$ whenever every integer less than $n$ and greater than or equal to $a$ belongs to $S$. Then, $S$ contains every integer greater than or equal to $a$.
\end{theorem}