Existence_of_an_object_with_a_property.tex

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\activitytitle{Showing existence of an object with a property}{}
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\overview{
Many proofs in mathematics require us to show that an object having a certain property exists.
Here are a few exercises of this type.
}

\prove{Let $a > 0$.
Show that there exists an integer $n$ with $\frac{1}{n} < a$.
Describe how to construct $n$ and then show that it works.}{0.9in}

\prove{Let $a$ and $b$ be real numbers with $a < b$.
Show that there exists a real number $c$ with $a < c < b$.
Describe how to construct $c$ and then show that it works.}{0.9in}

\prove{Let $a$ and $b$ be real numbers with $b - a > 1$.
Show that there exists an integer $n$ with $a < n < b$.
Describe how to construct $n$ and then show that it works.}{0.9in}

\prove{Let $a$ and $b$ be real numbers with $a < b$.
Show that there exists a rational number $r$ with $a < r < b$.
Describe how to construct $r$ and then show that it works.}{1in}

\prove{Suppose that $x = 4.32\overline{764}$, meaning that the decimal expansion has 764 repeating forever.
Show that $x$ is a rational number.
Start by explaining what needs to exist and what properties it needs.
Hint:  Look at 100000x - 100x.}{0.9in}

\prove{Suppose that $x$ is a real number with a repeating decimal expansion (of $d$ repeating digits starting after the $p$th decimal place).
Show that $x$ is a rational number.}{0in}

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