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\title{MATH3210 - Spring 2024 - Section 001}
\author{Homework 2}

\begin{document}

\maketitle

\begin{problem}
Consider the expression $x = .99999\dots$, which we think of as a decimal expansion. That is, if $x_n = \displaystyle\sum_{k=1}^n 9 \cdot 10^{-k}$,  we think of it as \[x = \lim_{n\to \infty} .9999\dots 9 = \lim_{n\to \infty} x_n.\] Show that $x = 1$. [{\it Hint}: First show that $x_n + 10^{-n} = 1$ by induction]
\end{problem}


\begin{problem}
Consider the following property for a set $A \subset \R$.

\begin{center}
(N) If $x \in A$, then there exists some $\ve > 0$ such that if $\abs{x -y} < \ve$, then $y \in A$.
\end{center}

Show that if $A$ is a bounded subset of $\R$ with property (N), then $\sup A \not\in A$.
\end{problem}

\begin{problem}
Show that if $(a_n)$ is increasing, then for every $N \in \N$, if $n \ge N$, $a_n \ge a_N$. [{\it Hint:} Use induction]
\end{problem}

\begin{problem}
Assume $(a_n)$ is increasing. Show that if $(a_n)$ is not bounded above if and only if $(a_n)$ diverges to $\infty$.
\end{problem}

\begin{problemstar}[Digit Expansions, Extra Credit]
Let $\Sigma_{10}$ denote the set of infinite sequences in the digits $\set{0,1,\dots,9}$. Denote an element of $\Sigma_{10}$ by $\sigma = (\sigma_1,\sigma_2,\dots)$, where each $\sigma_i \in \set{0,1,\dots,9}$. If $\sigma \in \Sigma$, let $(a(\sigma)_n)$ denote the sequence

\[ a(\sigma)_n = \sum_{k=1}^n \sigma_i \cdot 10^{-k}.\]

Show that $(a(\sigma)_n)$ converges. [{\it Hint}: Use Problem 1 and the Monotone convergence theorem]
\end{problemstar}
\end{document}


