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\def\myurl{http://www.math.toronto.edu/~drorbn/}

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Name (Last, First): $\underline{\hspace{3in}}$ \hfill Student ID: $\underline{\hspace{2in}}$

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{\small
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/classes/}{Classes}:
  \href{\myurl/classes/\#1415}{2014-15}:
  \href{\myurl/classes/15-475-ProblemSolving}{MAT 475 Problem Solving Seminar}:
  \hfill\url{http://drorbn.net/15-475}
}

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{\large\bf Quiz 8} on March 10, 2015: ``argue by contradiction'' and the Pigeonhole Principle. You have 30 minutes to solve the two problems below. Please write on both sides of the page. \hfill {\bf Good Luck!}

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\noindent{\bf Marking Comment.} At least for now, I have decided to simplify the management of this course and mark the quizzes myself, though at a delay of one week, in symbolic acknowledgement of the ongoing TA strike.

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\noindent{\bf Problem 1} (Larson's 2.6.10, expanded).
\begin{enumerate}
\item (4 points) Let $X$ be any real number. Prove that among the numbers
\[ X,\ 2X,\ \ldots,\ (n-1)X \]
there is at least one that differs from an integer by at most $1/n$.
\item (4 points) Let $\alpha$ be an irrational number. Prove that the set $\{\{n\alpha\}\colon n\in\bbZ\}$ is dense in the unit interval $[0,1]$, where for a real number $x$, $\{x\}$ denotes its ``fractional part'' --- the difference between $x$ and the largest integer $\lfloor x\rfloor$ smaller or equal to $x$.
\end{enumerate}

\noindent{\bf Problem 2} (5 points, Larson's 1.9.5, abbreviated). A set $S$ of rational
numbers is closed under addition and multiplication, and it is given that for
every $r\in\bbQ$ exactly one of the following is true: $r\in S$, $-r\in S$,
$r=0$. Prove that $S$ is the set of all positive rational numbers.

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