\documentclass[letterpaper,12pt]{article}

\usepackage{amsfonts,amssymb,amsmath,amsthm,fullpage,mathrsfs,enumitem}
%,txfont,pxfontss}
\usepackage{xcolor}

\pagestyle{empty}
\begin{document} 

\begin{center}\textbf{Problem Set 4 \hspace{.7in} Real Analysis 1 \hspace{.7in} Due 9/23/22} \end{center}

\noindent FTen of these problems will be graded, with each problem worth 2 points. Clear and complete justification is required for full credit. You are welcome to discuss these problems with anyone and everyone, but must write up your own final submission without reference to any sources other than the textbook and instructor.

\begin{enumerate}

\item If $\{a_n\}$ is a Cauchy sequence and $S=\{a_n|n \in \mathbb N \}$ is finite, then $\{a_n\}$ is constant from some point on.

\item Let $s_0$ be an accumulation point of $S$.  Then any neighborhood of $s_0$ contains at least one point of $S$ different from $s_0$ iff any neighborhood of $s_0$ contains infinitely many points of $S$.

\item Suppose that $\lim_{x \to \infty} f(x) = A$, where $f$ is a function with domain $D$.  Prove (directly from the definition) that if $k$ is a constant, $\lim_{x \to \infty} [k f(x) ] = k A $.

\item Suppose that $\lim_{x \to \infty} f(x) = A$, where $f$ is a function with domain $D$.  Prove (directly from the definition) that $ \lim_{x \to \infty} \left | f(x) \right | =  \left | \lim_{x \to \infty}  f(x) \right |$.

\item Suppose that $\lim_{x \to \infty} f(x) = A$ and $\lim_{x \to \infty} g(x) = B$, where $f$ and $g$ are functions with domain $D$.  Prove (directly from the definition) that $\lim_{x \to \infty} [f(x) \cdot g(x)] = A \cdot B$.

\item If a limit exists for $f$ as $x$ approaches $a$, then that limit is unique.

\item Suppose that $\lim_{x \to \infty} f(x) = A$ and $\lim_{x \to \infty} g(x) = B$, where $f$ and $g$ are functions with domain $D$. If $\forall x \in D, f(x) < g(x)$ then $A < B$.

\item Suppose that $\lim_{x \to \infty} f(x) = A$ and $\lim_{x \to \infty} g(x) = B$, where $f$ and $g$ are functions with domain $D$. If $\forall x \in D, f(x) \leq g(x)$ then $A \leq B$.

\item If $f:D \to \mathbb{R}$ and $\lim f(x)_{x \to a}$ exists, then $f$ is bounded.

\item If $f:D \to \mathbb{R}$ and $\lim f(x)_{x \to a}$ exists, then $f$ is bounded on some set $D_1$, with $D_1 \subseteq D$.

\end{enumerate}

\end{document}