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\item \textbf{Decimal Fractions}.
 \begin{enumerate}
 \item Prove that every rational number has either a terminating or repeating expansion. (Hint: long division.)
 \item In light of the last question, give me a clear way to generate an infinite decimal expansion that does not repeat, thus constructing an irrational number. Be sure I can tell how to generate each digit and also make an argument why it does not repeat.
 \item \label{series} Consider an infinite decimal expansion $0.d_1d_2d_3\dots$. Write it as a limit of a sequence of truncated (``cut-off'') decimal representations.
 \item Prove that the sequence in the last question is Cauchy. (Reminder: a sequence ${a_n}$ is \emph{Cauchy} if for any $\epsilon>0$, there is a natural number $N$ such that if $m,n > N$, then $|a_m - a_n| < \epsilon$.)
 \item (don't turn in) Remind yourself what a completion of a metric space is, and convince yourself that the real numbers are the completion of the rational numbers.
 \item Write $0.999\dots$ as the limit of a sequence of terminating decimals. Prove that the limit of this sequence is $1$. Yes, you'll have to use an $\epsilon-N$ argument.
 \end{enumerate}
 \item \textbf{Comparing Set Cardinalities}. Prove using the definition of ``same size'' and ``fits in'' in class that the following pairs of sets have the same size:
 \begin{enumerate}
 \item Prime numbers and whole numbers.
 \item Terminating decimals and repeating decimals (here you can count an infinitely repeating 0 as repeating).
 \end{enumerate}
 \end{enumerate}


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