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 \noindent \textbf{Big Problem 7: Which Is Bigger? }. Compare the cardinalities of the following sets:
 \begin{enumerate}

\item $\mathbb{N}$

\item $\mathbb{R}$

\item $\mathbb{R}^2$

\item $\mathbb{R}^n$
 \item set of functions from $\mathbb{N} \to \mathbb{N}$

\item set of functions from $\mathbb{R} \to \mathbb{R}$

\item algebraic numbers (i.e. the roots of finite degree polynomials with integer coefficients)

\item the Cantor Set

\item and the open interval $(0, 1)$
 \item set of all things you can describe in finite English sentences \\[.1in]
 \end{enumerate}

\noindent \textbf{Big Problem 8: Tulie Numbers}.
 Recall the Tulie Numbers class activity.
 \begin{enumerate}

\item Show that every rational number is dull, i.e. not endless.

\item Show that every dull number is rational.

\item Explain why the last two statements imply that a number has an infinite continued fraction representation if and only if it is irrational.

\item Convince me that $e$ is irrational by finding a pattern in its continued fraction expansion.\\[.1in]
 \end{enumerate}

\noindent \textbf{Big Problem 9: Irrational Numbers}.
 \begin{enumerate}
 \item Use an argument similar to Euclid's method to prove $\sqrt[n]{2}$ is irrational for any natural number $n > 1$.
 \item Use an argument similar to Euclid's method to prove that $\sqrt{p}$ is irrational for $p$ prime.
 \item Use the continued fractions method to prove the square root of $n^2+1$ is irrational for any natural number $n > 0$. (You can assume the results of Big Problem 8.)\\[.1in]
 \end{enumerate}

\noindent \textbf{Big Problem 10: Closed Form Formula for the Fibonacci Sequence}.

\begin{enumerate}

\item We call a sequence $A_n$ \emph{Fibonacci-esque} if it has the property $A_{n+2} = A_{n+1} + A_n$.

\item Find all geometric sequences that are Fibonacci-esque. That is, all numbers $C$ such that the sequence $A_n = kC_n$ is Fibonacci-esque (for constant $k$).

\item Prove linear combinations of Fibonacci-esque sequences have to be Fibonacci-esque.

\item Find a closed-form formula for the famous Fibonacci sequence ${1, 1, 2, 3, 5, 8, \dots}$ by finding a sum of Fibonacci-esque geometric sequences that has the famous values.
 \end{enumerate}
 
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