\documentclass[letterpaper,12pt]{article}

\usepackage{amsfonts,amssymb,amsmath,amsthm,txfonts,pxfonts,mathrsfs,fullpage,enumitem}
\usepackage{xcolor}

\pagestyle{empty}
\begin{document} 

\begin{center}\textbf{Problem Set 8 \hspace{.7in} Real Analysis 1 \hspace{.7in} Due 11/20/20} \end{center}

\noindent Four of these problems will be graded, with each problem worth 5 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 Let $f$ be continuous at $x=c \in [a,b]$.  Let ${x_n}$ be a sequence in $[a,b]$ converging to $c$.  Then the sequence $\{f(x_n)\}$ converges to $f(c)$.
\item Let $f(x) = x$, and let $\mathscr{P}$ be a regular partition of $[a,b]$ with $n$ subdivisions.  Evaluate $U(\mathscr{P},f)$ and $L(\mathscr{P},f)$, and find their limits as $n$ approaches $\infty$.
\item Suppose that a function $f:[a,b] \to \mathbb{R}$ is bounded and $\mathscr{P} = {x_0,x_1,x_2,...,x_n}$ is a partition of $[a,b]$.  Then there exist $m, M \in \mathbb{R}$ such that
$$m(b-a)\leq L(\mathscr{P},f) \leq S(\mathscr{P},f) \leq U(\mathscr{P},f) \leq M(b-a)$$
\item Suppose that a function $f:[a,b] \to \mathbb{R}$ is bounded and $\mathscr{P}$ and $\mathscr{Q}$ are partitions of $[a,b]$.  Show that if $\mathscr{P} \subseteq \mathscr{Q}$, then $U(\mathscr{Q} , f ) \leq U( \mathscr{P}, f )$.

\end{enumerate}

\end{document}