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\begin{document}

\title{Homework 1}


\author{Marius Ionescu}
\maketitle
\begin{problem}[Exercise 1.2.1 on page 11]
  \begin{enumerate}
  \item Prove that $\sqrt{3}$ is irrational. Does a similar argument
    work to show $\sqrt{6}$ is irrational?
  \item Where does the proof of Theorem 1.1.1 break down if we try to
    use it to prove that $\sqrt{4}$ is irrational?
  \end{enumerate}
\end{problem}
%\begin{proof}
  
%\end{proof}
\begin{problem}[Exercise 1.5.4 on page 31]
  Let $S$ be the set consisting of all sequences of 0's and
  1's. Observe that $S$ is not a particular sequence, but rather a
  large set whose elements are sequences, namely
  \[
  S=\{(a_1,a_2,a_3,\dots )\,:\, a_n=0\text{ or }1\}
  \]
\end{problem}
%\begin{proof}
  
%\end{proof}
\begin{problem}[Exercise  1.3.3 a) on page 17]
  Let $A$ be bounded below, and define $B = \{b\in\R : 
b$ is a lower bound for $A\}$.  Show that $\sup B = \inf A$.
\end{problem}
\begin{problem}[Exercise 1.3.4 on page 18]
  Assume that $A$ and $B$ are nonempty, bounded 
above, and satisfy $B\subseteq A$.  Show that $\sup B \leq \sup A$.
\end{problem}
\begin{problem}[Exercise 1.3.6 in page 18]
  Compute, without proofs, the suprema and infima of the
following sets:
\begin{itemize}
\item[(a)] $\{n\in\N : n^2<10\}$.
\item[(b)] $\{n/(m+n) : m,n\in\N\}$.
\item[(c)] $\{n/(2n+1) : n\in\N\}$.
\item[(d)] $\{n/m : m,n\in\N$ with $m+n\leq10\}$.
\end{itemize}
\end{problem}
\begin{problem}[Exercise 1.3.7 on page 18]
  Prove that if $a$ is an upper bound for $A$, and if
$a$ is also an element of $A$, then it must be that $a=\sup A$.
\end{problem}
\end{document}
