41 lines
905 B
TeX
41 lines
905 B
TeX
\section{Proofs by Induction}
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\definition{}
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The last proof technique we'll discuss in this handout is \textit{induction.} \par
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This is particularly useful when we have a \say{countable} variable, usually an integer. \par
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\vspace{2mm}
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A proof by induction consists of two parts: a \textit{base case} and a \textit{inductive step}. \par
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\vfill
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Note that although induction is a powerful proof technique, it usually leads to uninteresting results. \par
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If we prove a statement using induction, we conclude that it is true---but we get very little insight on
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\textit{why} that is.
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\vspace{2mm}
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Alternative proofs are take a bit more work than inductive proofs, but they are much more valuable. \par
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For example, consider the following proof of X:
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\makeatletter
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\@makeORMCbox{tmpbox}{Alternative Proof}{ogrape!10!white}{ogrape}
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\makeatother
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\begin{tmpbox}
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sdfasdf
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\end{tmpbox}
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\pagebreak |