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{\end{exmp}}

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\pagestyle{fancy} \lhead{{\sf Homework 6 $|$ Due February 24 (Tuesday)
}} \rhead{\thepage} \cfoot{{\sf Math 3190 $|$ Introduction to Proof $|$
    Spring 2026 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-4mm}

%%------------------------------------------------------------------------


\begin{exercise} % HW 6, Problem 1
Prove that the sequence $\big\{\frac{2n^2+2n+1}{3n^2}\big\}$ converges to $\frac{2}{3}$.
\[
\centering
\begin{tikzpicture}[scale=.75,yscale=1.5]
    \def\N{13}
    \def\eps{.6}
    \tikzstyle{v} = [circle, draw, fill=black,inner sep=0pt, minimum size=1mm]
    %%
    % ---- Define the sequence formula here ----
    \newcommand{\an}[1]{((2*(#1)^2+2*(#1)+1)/(3*(#1)^2))}
    %\newcommand{\an}[1]{(1-1/(#1))*(-1)^(#1)}
    %\newcommand{\an}[1]{(#1)*((-1)^(#1))/(#1+4)}
    %\draw[black!10,fill=black!10] (0,1-\eps) rectangle (\N+.8,1+\eps);
    \draw[<->] (0,-.7) to (0,2.5);
    \draw[<->] (-1,0) to (\N+.8,0);
    \draw[very thin,gray,dashed] (0,.667) to (\N+.8,.667);

    % ---- Automatically generate N points ----
    \foreach \i in {1,...,\N} {
        \pgfmathsetmacro{\y}{\an{\i}}
        \node[v] at (\i,\y) {};
        \draw[thin] (\i,-.1) to (\i,.1);
    }    
    \node at (-.6,2) {\footnotesize $2$};
    \draw (-.15,2) to (.15,2);
    \node at (-.6,1) {\footnotesize $1$};
    \draw (-.15,1) to (.15,1);

    % ---- labels on the x-axis ----
    \foreach \i in {5,10} {
        \node at (\i,-.35) {\footnotesize $\i$};
        \draw[thick] (\i,-.15) to (\i,.15);
    }    
\end{tikzpicture}
\]
\end{exercise}


\begin{exercise} % HW 6, Problem 2 
Prove that the sequence $\big\{1-\frac{1}{2^n}\big\}$ converges to $1$.
\[
\centering
\begin{tikzpicture}[scale=.75,yscale=1.5]
    \def\N{13}
    \def\eps{.6}
    \tikzstyle{v} = [circle, draw, fill=black,inner sep=0pt, minimum size=1mm]
    %%
    % ---- Define the sequence formula here ----
    \newcommand{\an}[1]{(1-1/(2^(#1)))}
    %\newcommand{\an}[1]{(1-1/(#1))*(-1)^(#1)}
    %\newcommand{\an}[1]{(#1)*((-1)^(#1))/(#1+4)}
    %\draw[black!10,fill=black!10] (0,1-\eps) rectangle (\N+.8,1+\eps);
    \draw[<->] (0,-1.75) to (0,1.75);
    \draw[<->] (-1,0) to (\N+.8,0);
    \draw[very thin,gray,dashed] (0,1) to (\N+.8,1);
    \draw[very thin,gray,dashed] (0,-1) to (\N+.8,-1);

    % ---- Automatically generate N points ----
    \foreach \i in {1,...,\N} {
        \pgfmathsetmacro{\y}{\an{\i}}
        \node[v] at (\i,\y) {};
        \draw[thin] (\i,-.1) to (\i,.1);
    }    
    \node at (-.6,1) {\footnotesize $1$};
    \node at (-.6,-1) {\footnotesize $-1$};
    \draw (-.15,1) to (.15,1); 
    \draw (-.15,1-\eps) to (.15,1-\eps); 

    % ---- labels on the x-axis ----
    \foreach \i in {5,10} {
        \node at (\i,-.35) {\footnotesize $\i$};
        \draw[thick] (\i,-.15) to (\i,.15);
    }    
\end{tikzpicture}
\]
\end{exercise}


\begin{exercise} % HW 6, Problem 3
Consider the sequence $\big\{(-1)^n\big(1+\frac{1}{\sqrt{n}}\big)\big\}$. The first $20$ terms are shown below.
\[
\centering
\begin{tikzpicture}[scale=.6,yscale=1.5]
    \def\N{20}
    \def\eps{.6}
    \tikzstyle{v} = [circle, draw, fill=black,inner sep=0pt, minimum size=1mm]
    %%
    % ---- Define the sequence formula here ----
    \newcommand{\an}[1]{(1+1/(sqrt(#1)))*(-1)^(#1)}
    %\newcommand{\an}[1]{(1-1/(#1))*(-1)^(#1)}
    %\newcommand{\an}[1]{(#1)*((-1)^(#1))/(#1+4)}
    %\draw[black!10,fill=black!10] (0,1-\eps) rectangle (\N+.8,1+\eps);
    \draw[<->] (0,-2) to (0,2);
    \draw[<->] (-1,0) to (\N+.8,0);
    \draw[very thin,gray,dashed] (0,1) to (\N+.8,1);
    \draw[very thin,gray,dashed] (0,-1) to (\N+.8,-1);

    % ---- Automatically generate N points ----
    \foreach \i in {1,...,\N} {
        \pgfmathsetmacro{\y}{\an{\i}}
        \node[v] at (\i,\y) {};
        \draw[thin] (\i,-.1) to (\i,.1);
    }    
    \node at (-.6,1) {\footnotesize $1$};
    \node at (-.6,-1) {\footnotesize $-1$};
    \draw (-.15,1) to (.15,1); 
    \draw (-.15,1-\eps) to (.15,1-\eps); 

    % ---- labels on the x-axis ----
    \foreach \i in {5,10,15,20} {
        \node at (\i,-.35) {\footnotesize $\i$};
        \draw[thick] (\i,-.15) to (\i,.15);
    }    
\end{tikzpicture}
\]
We wish to show that this sequence diverges. For sake of contradiction, suppose that it converges to a limit $L\in\R$. Let $\epsilon=1$, and arrive at a contradiction. [\emph{Hint}: See Example 13.9 in Hammack for this technique used on a different sequence.]

\end{exercise}

%\begin{solution} 
    %% Reminder to cite AI usage, if applicable. 
    % Put it immediately after \begin{solution} (on the same line)
    %\AIpolish
    %\AImajor
    %\AIminor
    %\AIpolishmajor
    %\AIpolishminor
    %
    % To do.
%\end{solution}

\smallskip

%%------------------------------------------------------------------------

\begin{exercise} % HW 6, Problem 4 (Taylor 1.17)
  For each set $X$ below, determine which of the following statements are true. 
\begin{multicols}{3}
\begin{enumerate}[label=(\roman*)]
    \item $a\in X$
    \item $\{a\}\in X$
    \item $a\subseteq X$
    \item $\{a\}\subseteq X$
    \item $\{a,b\}\in X$
    \item $\{a,b\}\subseteq X$
    \item $\big\{\{a,b\}\big\}\in X$
    \item $\big\{\{a,b\}\big\}\subseteq X$
    \item $\big\{\{a,b,c,d\}\big\}\subseteq X$
\end{enumerate}  
\end{multicols}

\begin{multicols}{2}
\begin{enumerate}%[label=(\roman*)]
    \item $X=\big\{a,b,\{c,d\}\big\}$
    \item $X=\big\{c,d,\{a,b\}\big\}$
    \item $X=\big\{a,b,\{a,b\}\big\}$
    \item $X=\big\{a,\{a\},b,\{b\}\big\}$
    \item $X=\big\{\{a,b\}\big\}$
    \item $X=\big\{\{a\},\{b\},\{a,b\}\big\}$
    \item $X=\Big\{\big\{\{a\}\big\},\{b\},\big\{\{a\},b\big\}\Big\}$
    \item $X=\Big\{\big\{a\big\},\big\{a,b\big\},\big\{\{a\}\big\},\big\{\{a,b\}\big\}\Big\}$
\end{enumerate}  
\end{multicols}
\end{exercise}

%\begin{solution} 
    % To do.
%\end{solution}

\smallskip

%%------------------------------------------------------------------------

\begin{exercise} % HW 6, Problem 5 
Consider the following set of matrices:
\[
   R=\Bigg\{\underbrace{\begin{bmatrix}0&0\\0&0\end{bmatrix}}_{0},\;
   \underbrace{\begin{bmatrix}1&0\\0&1\end{bmatrix}}_{a},\;
   \underbrace{\begin{bmatrix}2&0\\0&2\end{bmatrix}}_{b},\;
   \underbrace{\begin{bmatrix}3&0\\0&3\end{bmatrix}}_{c}\Bigg\}.
\]
This is an example of a \emph{set with structure}. In particular, if we are working modulo $4$ (i.e., $4\equiv 0$, $5\equiv 1$, $6\equiv 2$, etc.), then we can add, subtract, and multiply  these matrices by each other, we stay within the set---it is called a \emph{ring}. We can describe this by addition, subtraction, and multiplication tables, like shown below: 
\[
\renewcommand{\arraystretch}{1}
\begin{array}{c|cccc} 
  + & 0 & a & b & c \\ \hline
  0 & 0 & a & b & c \\ 
  a & a & b & c & 0 \\ 
  b & b & c & 0 & a \\ 
  c & c & 0 & a & b
\end{array}
\hspace{15mm}
\begin{array}{c|cccc}
  - & 0 & a & b & c \\ \hline
  0 & 0 & c & b & a \\ 
  a & a & 0 & c & b \\ 
  b & b & a & 0 & c \\ 
  c & c & b & a & 0
\end{array}
\hspace{15mm}
\begin{array}{c|cccc} 
  \times & 0 & a & b & c \\ \hline
  0 & 0 & 0 & 0 & 0 \\ 
  a & 0 & a & b & c \\ 
  b & 0 & b & 0 & b \\ 
  c & 0 & c & b & a
\end{array}
\]
\begin{enumerate}[label=(\alph*)]
    \item Construct these three tables for each of the following two sets of matrices, where this time, everything is modulo $2$ (i.e., $2\equiv 0$, $3\equiv 1$, $4\equiv 0$, etc.).
\[\hspace*{-5mm}
   F=\Bigg\{\underbrace{\begin{bmatrix}0&0\\0&0\end{bmatrix}}_{0},\;
   \underbrace{\begin{bmatrix}1&0\\0&1\end{bmatrix}}_{a},\;
   \underbrace{\begin{bmatrix}1&1\\1&0\end{bmatrix}}_{b},\;
   \underbrace{\begin{bmatrix}0&1\\1&1\end{bmatrix}}_{c}\Bigg\},\quad
   V=\Bigg\{\underbrace{\begin{bmatrix}0&0\\0&0\end{bmatrix}}_{0},\;
      \underbrace{\begin{bmatrix}1&0\\0&1\end{bmatrix}}_{a},\;
      \underbrace{\begin{bmatrix}1&0\\0&0\end{bmatrix}}_{b},\;
      \underbrace{\begin{bmatrix}0&0\\0&1\end{bmatrix}}_{c}
      \Bigg\}.
\]
\item In each of these sets $S$, the matrix $a$ is the (multiplicative) \emph{identity}, because $ax=x=xa$ for all other matrices $x\in S$. A matrix $x$ has a (multiplicative) \emph{inverse} $y$ if $xy=a=yx$. For each of these sets ($R$, $F$, and $V$), find the inverse of each nonzero element $a$, $b$, and $c$, if it has one.
\end{enumerate}
\end{exercise}

%\begin{solution} 
    % To do.
%\end{solution}

%%------------------------------------------------------------------------

\end{document}

