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\begin{center}\textbf{Exam 1 \hspace{.7in} Real Analysis 1 \hspace{.7in} 9/30/16} \end{center}

\noindent Each problem is worth 10 points. Show adequate justification for full credit. Don't panic.

\begin{enumerate}
\item State the definition of a function $f(x)$ converging as $x$ approaches $a$.

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\item a)  State the definition of an accumulation point.\\
\vspace{3in}\\
b)  Give an example of a set which is infinite but has no accumulation points.

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\item a)  State the definition of a Cauchy sequence.\\
\vspace{3in}\\
b)  Give an example of a sequence which is Cauchy.

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\item a)  Give an example of a set which is infinite and bounded, and which has a maximum element.\\
\vspace{4in}\\
b)  Give an example of a set which is infinite and bounded, and which does not have a maximum element.

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\item a)  State the Bolzano-Weierstrass Theorem for Sets.\\
\vspace{3in}\\
b)  State the Triangle Inequality.

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\item a)  State the Sum Rule for limits of sequences.\\
\vspace{3in}\\
b)  State the Quotient Rule for limits of sequences.

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\item State and prove the Product Rule for limits of products of functions as $x$ approaches $a$.

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\item If a sequence $\{a_n\}$ diverges to $+\infty$ and $a_n \leq b_n$ for all $n \geq n_1$, then the sequence $\{b_n\}$ must also diverge to $+\infty$.

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\item a)  Prove or give a counterexample: If a sequence $\{a_n\}$ is convergent, then it is eventually increasing and bounded.\\
\vspace{4in}\\
b)  Prove or give a counterexample: If a sequence $\{a_n\}$ is eventually increasing and bounded, then it is convergent.

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\item If $f:D \to \mathbb{R}$ and $\lim_{x \to a} f(x)$ exists, then $f$ is bounded on some set $D_1$, with $D_1 \subseteq D$ and $a$ an accumulation point of $D_1$.
\end{enumerate}

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