\section{Proofs by Induction}

\definition{}
The last proof technique we'll discuss in this handout is \textit{induction.} \par
This is particularly useful when we have a \say{countable} variable, usually an integer. \par

\vspace{2mm}

A proof by induction consists of two parts: a \textit{base case} and a \textit{inductive step}. \par











\vfill

Note that although induction is a powerful proof technique, it usually leads to uninteresting results. \par
If we prove a statement using induction, we conclude that it is true---but we get very little insight on
\textit{why} that is. 

\vspace{2mm}

Alternative proofs are take a bit more work than inductive proofs, but they are much more valuable. \par
For example, consider the following proof of X:

\makeatletter
\@makeORMCbox{tmpbox}{Alternative Proof}{ogrape!10!white}{ogrape}
\makeatother

\begin{tmpbox}
	sdfasdf
\end{tmpbox}


\pagebreak