constructions_to_show_existence.tex

\yourname

\activitytitle{Construction of an object with a property}{}
\vspace*{-0.2in}

\overview{
Many proofs in mathematics require us to show that an object having a certain property exists.
In many cases, you can tell exactly how to make, or construct, the desired object.
}

\prove{Let $x>0$ be a real number.

\balist{0.2in}
\item Construct a real number $a$ with $0 < a < x$.
That is, write a formula to compute $a$ in terms of $x$, then check that $0 < a < x$.

\vspace*{0.1in}

\item When $x = 0.0006$, what number does your formula produce for $a$?

\item Construct real numbers $a$ and $b$ with $0 < a < b < x$.

\vspace*{0.1in}

\item When $x = 0.0006$, what numbers does your formula produce for $a$ and $b$?

\item Find two very different ways to construct a real number $c$ satisfying $x < c$.
\elist
}{0.2in}

\prove{Given a real number $x$, describe a procedure that results in one integer $n$ with $n > x$.

\vspace{0.5in}
\noindent
What result does your procedure give for $x = 13.1$?  \hfill $x = 18$?  \hfill $x = -22.2$?
}{0.2in}

\prove{
\balist{0.1in}
\item Find the smallest integer $n$ with $n^2 > 17$.

\item Find the smallest integer $n$ with $n^2 > 177$.

\item Describe a procedure for finding the smallest integer $n$ with $n^2 > 1777$.

\vspace*{0.1in}

\item Given an integer $k>1$, describe a short procedure to find the smallest integer $n$ with $n^2 > k$.

\vspace*{0.1in}

\item Describe a procedure for finding the largest integer $n$ with $n^2 \leq 1777$.

\elist
}{0.2in}

\prove{Given a real number $a$, construct an integer $n$ with $a < n \leq a+1$.
You may wish to consider two cases:  Case 1, suppose $a$ is an integer.  Case 2, suppose $a$ is not an integer.

\vspace{0.9in}
\noindent
What value of $n$ does your procedure give when $a = 13.1$?  \hfill $a = 18$?  \hfill $a = -22.2$?
}{0.0in}

% ==========================================================
\pagebreak

\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 work with inequalities to show that it works.}{0.9in}

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

\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.
{\bf 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.}{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.}{0.0in}


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