

%This is how you make a comment.

\documentclass{article}

%-------Packages---------
\usepackage{amssymb,amsfonts,amsmath}
\usepackage{hyperref}
\usepackage[]{amsthm} %lets us use \begin{proof}

%--------Theorem Environments--------
%theoremstyle{plain} --- default
\newtheorem{thm}{Theorem}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{lem}[thm]{Lemma}

\newtheorem{quest}[thm]{Question}

\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}

\newtheorem{exer}{Exercise}



\bibliographystyle{plain}

%--------Meta Data: Fill in your info------
\title{Math 3100 Homework Assignment 3 - Due Wednesday February 17th}

\author{Your name here}


\date{\today}

\begin{document}

%\begin{abstract}

%This is where you would add an abstract

%\end{abstract}

\maketitle

%\tableofcontents


\begin{exer}
Suppose $0 \leq a \leq b$. Determine for what values of $a$ and $b$ the sequence $\{ (a^n + b^n)^{1/n} \}_{n = 1}^\infty$ has a limit. When it does, compute the limit and justify your answer. 
\end{exer}


\noindent\makebox[\linewidth]{\rule{\textwidth}{0.4pt}}


\begin{exer}
Let $\{ a_n \}_{n = 1}^\infty$ be a sequence of real numbers and let $x$ be any accumulation point of the set of values $\{ a_n \mid n \in \mathbb Z_+ \}$. Prove that there is some subsequence that converges to $x$. 
\end{exer}





\noindent\makebox[\linewidth]{\rule{\textwidth}{0.4pt}}

\begin{exer}
Fix $x \in \mathbb R$. Prove that there exists a sequence $\{ a_k \}_{k = 1}^\infty$ taking values in the set $\{ -1, 1 \}$ such that the sequence $\{ \sum_{k = 1}^n \frac{a_k}{k} \}_{n = 1}^\infty$ converges to $x$. 
\end{exer}




\noindent\makebox[\linewidth]{\rule{\textwidth}{0.4pt}}

\begin{exer}
Recall that any number $x \in [0, 1)$ has a unique representation in base-2 so that it does not end in a string of $1$'s, i.e., for each $x \in [0, 1)$ there is a unique sequence $\{ a_n \}_{n = 1}^\infty$ taking values in the set $\{ 0, 1 \}$ such that the sequence is not eventually equal to $1$ and such that $x$ has the binary expansion
\[
x = 0.a_1a_2a_3 \cdots,
\] 
or equivalently $x = \lim_{n \to \infty} \sum_{k = 1}^n \frac{a_k}{2^k}$. Consider the function $f: [0, 1) \to \mathbb R$ that takes the binary representation of $x$ and reinterprets it as a decimal representation, e.g., if $x$ is as before then $f(x) = \lim_{n \to \infty} \sum_{k = 1}^n \frac{a_k}{10^k}$. Determine all values in $[0, 1]$ where this function has a limit. Justify your answer.  
\end{exer}







%\begin{thebibliography}{9}



%\bibitem{halmosmath}
%Paul R. Halmos
%How to write mathematics.
%\url{https://sites.math.washington.edu/~lind/Resources/Halmos.pdf}

%\bibitem{knuthmath}
%Donald E. Knuth, Tracy Larrabee and Paul M. Roberts
%Mathematical Writing.
%\url{https://jmlr.csail.mit.edu/reviewing-papers/knuth_mathematical_writing.pdf}

%\end{thebibliography}

\end{document}

