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\begin{document}

\hbox to \hsize{\textbf{Math 331\hfill Homework 1}}
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\medskip

\centerline{\textit{This homework is due by the end of the day on Wednesday, Sept. 2}}
\centerline{\textit{Many of the problems are from the exercises for Sections 1.1 and 1.2.}}

\bigskip

\begin{problem}
Prove by contradiction that $\log_{10}2$ is irrational.  (Hint: Suppose that $\log_{10}2=p/q$ 
for some integers $p$ and $q$, where $q\ne 0$.  Show that this would imply $2^q=10^p$, 
and apply the Fundamental Theorem of Arithmetic.)
\end{problem}
\begin{answer}
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\begin{problem}
Prove that if $a$ is irrational, then $\sqrt a$ is also irrational.
\end{problem}
\begin{answer}
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\begin{problem}
Determine whether each set is bounded above and if so find its least upper bound: 
$A=\{1-\frac{1}{n}\,|\,n\in\N\}$; $B=\{\frac{n+1}{n}\,|\,n\in\N\}$; $C=[2,9)$;\vadjust{\smallskip}\\
$D=\{1,1+\frac{1}{2},1+\frac{1}{2}+\frac{1}{4},1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8},1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16},\dots\}$;\vadjust{\smallskip}\\
$E=\{1,1+\frac{1}{2},1+\frac{1}{2}+\frac{1}{3},1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4},1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5},\dots\}$. 
\end{problem}
\begin{answer}
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\begin{problem}
Let $\eps$ be any positive real number.  Show that there is a natural number, $n$,
such that $\frac{1}{n}<\eps$.  (Hint: Apply the Archimedian property of $\R$ to $\frac{1}{\eps}$.)
\end{problem}
\begin{answer}
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\begin{problem}
Consider two sets of real numbers $A=\{a_1,a_2,a_3,\dots\}$ and $B=\{b_1,b_2,b_3,\dots\}$.
Let $C$ be the set $C=\{a_1+b_1,a_2+b_2,a_3+b_3,\dots\}$.
    \begin{enumerate}
    \item[(a)] Suppose that $\mu_1$ is an upper bound for $A$ and $\mu_2$ is an upper bound
        for $B$.  Show that $\mu_1+\mu_2$ is an upper bound for $C$.
    \item[(b)] Now suppose that $\lambda_1$ is the least upper bound for $A$ and
        $\lambda_2$ is the least upper bound for $B$.  Give an example to show that
        $\lambda_1+\lambda_2$ is not necessarily the least upper bound of $C$.
    \end{enumerate}
\end{problem}
\begin{answer}
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\begin{problem}
Let $A$ and $B$ be arbitrary non-empty, bounded-above sets of real numbers.  Define
$C=\{a+b\,|\,a\in A$ and $b\in B\}$.  [Compare this to the previous problem,
where $C$ contains only the sums of corresponding elements of $A$ and $B$;
the $C$ in this problem contains many elements that are not in the $C$ in
the preceding problem.]
    \begin{enumerate}
    \item[(a)] Suppose that $\lambda_1$ is the least upper bound for $A$ and 
        $\lambda_2$ is the least upper bound for $B$.  Let $\lambda=\lambda_1+\lambda_2$.
        Show that $\lambda$ is an upper bound for $C$.
    \item[(b)] Now, show that $\lambda$ is the least upper bound for $C$. (Hint: Consider the
        last theorem in the third reading guide.  Let
        $\eps>0$.  Explain why there is a $a_o\in A$ such that $a_o> \lambda_1-\frac{\eps}{2}$
        and a $b_o\in B$ such that $b_o>\lambda_2-\frac{\eps}{2}$.  Use this to show
        $a_o+b_o>\lambda-\eps$, and conclude that $\lambda$ is the least upper bound for $C$.)
    \end{enumerate}
\end{problem}
\begin{answer}
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\begin{problem}
Let $X$ be a non-empty, bounded-below subset of $\R$, and let $\mu$ be a lower bound for $\R$.
Show that $\mu$ is the
greatest lower bound of $X$ if and only if for every $\eps>0$, there is
an $x\in X$ such that $x<\mu+\eps$.  (Hint: Compare to the last theorem in
the third reading guide.)
\end{problem}
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\begin{problem}
Prove that the intersection of two Dedekind cuts is again a Dedekind cut.
\end{problem}
\begin{answer}
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\end{answer}



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