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\noindent
\parbox{2in}{\bf Math 3220}
\hfill {\Large \bf Test \#1} \hfill
\parbox{2in}{\bf \hfill February $13^\mathrm{th}$\!,\  2019}

\vspace{0.2in}

\noindent {\large\bf Name:} \underline{\hskip 3.0 truein}
\hfill {\bf Be sure to show your work!}

\vspace{0.1in}


\noindent {\bf\large 1. (12 points)} Definitions

\begin{enumerate}[(a)]

\item Give the definition of a {\it Cauchy} sequence for a sequence $\{ x_n \}_{n=1}^\infty$.

\vfill

\item Let $X \subseteq \mathbb{R}$ be a non-empty set. Give the definition of $\mathrm{inf}(X)$.

\vfill

\end{enumerate}


\noindent {\bf\large 2. (25 points)} Determine which of the following statements are true or false. Circle your answer.

\begin{description}

\item[(a) \quad True \quad / \quad False \ \  ] $\mathbb{Q}$ (the rational numbers) and $\mathbb{R}-\mathbb{Q}$ (the irrational numbers) have the same cardinality.

\vspace{0.1in}

\item[(b) \quad True \quad / \quad False \ \  ] The set $A = \{ r \in \mathbb{Q} \;|\;  0 < r < \sqrt{2} \}$ is countably infinite.

\vspace{0.1in}

\item[(c) \quad True \quad / \quad False \ \  ] If $|x-3|>1$, then $x>4$.

\vspace{0.1in}

\item[(d) \quad True \quad / \quad False \ \  ] If $\{ x_n \}_{n=1}^\infty$ is a Cauchy sequence, then every subsequence of $\{ x_n \}_{n=1}^\infty$ converges.

\vspace{0.1in}

\item[(e) \quad True \quad / \quad False \ \  ] Convergent sequences are bounded.

\vspace{0.1in}

\item[(f) \quad True \quad / \quad False \ \  ] Bounded monotone sequences converge.

\vspace{0.1in}

\item[(g) \quad True \quad / \quad False \ \  ] If $X \subseteq \mathbb{R}$ is a non-empty set, then we can always conclude that $\mathrm{inf}(X) < \mathrm{sup}(X)$. 

\vspace{0.1in}

\item[(h) \quad True \quad / \quad False \ \  ] If $\{a_n \}_{n=1}^\infty$ converges and $\{b_n\}_{n=1}^\infty$ diverges, then $\{a_n + b_n\}_{n=1}^\infty$ diverges. 

\vspace{0.1in}

\item[(i) \quad True \quad / \quad False \ \  ] If $a_n \to A$ and $b_n \to B$, then $a_n/b_n \to A/B$. 

\vspace{0.1in}

\item[(j) \quad True \quad / \quad False \ \  ] If $\displaystyle \lim_{n \to \infty} a_n = -\infty$, then $\displaystyle \lim_{n \to \infty} 1/a_n = 0$.


\end{description}

\newpage

\noindent {\bf\large 3. (14 points)} Examples requested.

\begin{enumerate}[(a)]
\item Give an example of a set $A \subseteq \mathbb{R}$ such that $\mathrm{inf}(A)=-4 \not\in A$ and $\mathrm{sup}(A)=\infty$.

\vfill

\item Give an example of a {\it non-monotonic} sequence which converges to $5$.

\vfill

\end{enumerate}

\noindent {\bf\large 4. (15 points)} Let $S$ be a non-empty set of {\it positive} real numbers. Moreover, assume $S$ is bounded above. 
First, why does $\alpha = \mathrm{sup}(S)$ exist? 
Next, show that $\mathrm{sup}(S^2) \leq \alpha^2$ where $S^2 = \{ s^2 \;|\; s \in S\}$.

\vfill

\vfill

\newpage

\noindent {\bf\large 5. (12 points)} List off the first 4 terms in the sequence $\left\{ \dfrac{(-1)^nn+1}{n} \right\}_{n=1}^\infty$. Then show it diverges.

\vfill

\noindent {\bf\large 6. (22 points)} Use the {\it definition of convergence} in the following proofs. Don't appeal to ``big theorems''.

\begin{enumerate}[(a)]

\item Show that $\left\{ \dfrac{2n+3}{n^2+1} \right\}_{n=1}^\infty$ converges.

\vfill

\item Suppose $\{a_n\}_{n=1}^\infty$ and $\{b_n\}_{n=1}^\infty$ converge. Show that $\{a_n+b_n\}_{n=1}^\infty$ converges.

\vfill

\end{enumerate}




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