\documentclass{article}

\title{\textbf{Homework 2}}
%\author{Enrique Trevi\~no}

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\begin{document}

\maketitle
\normalsize
\section{Chapter 2}
\begin{problem}(\textbf{Exercise 2.2.1})\\
Prove that $f(x) = \sum_{n=1}^{\infty} ((nx))/n^2$ converges uniformly. Prove that the interchange of limits in Equation (2.9) is allowed.
\end{problem}
\begin{sol}
\end{sol}

For exercises 2.2.4 to 2.2.9 use $$\mathcal{F}(x) = \sum_{n=1}^{\infty} \frac{\cos(n! x)}{n!}.$$
\begin{problem}(\textbf{Exercise 2.2.4})\\
Show that $\mathcal F$ is a uniformly convergent series of continuous functions and therefore is continuous.
\end{problem}
\begin{sol}
\end{sol}

We shall show that $$\lim_{h\rightarrow\infty} \frac{\mathcal{F}(x+h) - \mathcal{F}(x)}{h}$$ does not exist at any $x$, and therefore $\mathcal{F}$ is nowhere differentiable. Fix an $\varepsilon >0$ and, for each $N \in \mathbb{N}$, define $h = \varepsilon/N!$. The variable $h$ depends on $N$.

\begin{problem}(\textbf{Exercise 2.2.5})\\
Show that for every positive integer $n$,
\begin{equation}\label{2.12}
\frac{\cos(n! (x+h) ) - \cos(n! x)}{n! h} = -\sin(n! x) - \frac{h}{2} n!\cos(n!c_n)
\end{equation}
for some $c_n$ between $x$ and $x+h$.
for some
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 2.2.6})\\
Show that $$\sum_{n=1}^{N-1} \frac{n!}{N!} \le \frac{2}{N},$$
and therefore
\begin{equation}\label{2.13}
\left|\sum_{n=1}^{N-1} \frac{h}{2} n! \cos(n! c_n)\right| \le \frac{\varepsilon}{2}\sum_{n=1}^{N-1}\frac{n!}{N!}\le \frac{\varepsilon}{N}.
\end{equation}
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 2.2.7})\\
Justify the equality
\begin{align*}
\frac{1}{h}\left(\sum_{n=N+1}^{\infty} \frac{\cos(n!(x+h))}{n!} - \sum_{n=N+1}^{\infty} \frac{\cos(n!x)}{n!}\right) = \sum_{n=N+1}^{\infty}\frac{\cos(n!(x+h))-\cos(n! x)}{n! h}.
\end{align*}
Justify each inequality,
\begin{align}\label{2.14}
\left|\sum_{n=N+1}^{\infty}\frac{\cos(n!(x+h))-\cos(n! x)}{n! h}\right| &\le \frac{2}{N! h}\left(\frac{1}{N+1} + \frac{1}{(N+1)(N+2)} + \frac{1}{(N+1)(N+2)(N+3)}+\ldots\right)\notag\\
&\le \frac{4}{\varepsilon N}.
\end{align}
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 2.2.8})\\
Using Equations \eqref{2.12}, \eqref{2.13}, and \eqref{2.14} we see that
\begin{align}\label{2.15}
\frac{\mathcal{F}(x+h) - \mathcal{F}(x)}{h} &= -\left(\sum_{n=1}^{N-1}\sin(n! x)\right) + \frac{\cos(N!(x+h)) - \cos(N! x)}{N! h} + E(\varepsilon, N)\notag\\
&= -\left(\sum_{n=1}^{N-1}\sin(n! x)\right) + \frac{\cos(N!x+\varepsilon)) - \cos(N! x)}{N! h} + E(\varepsilon, N),
\end{align}
where $|E(\varepsilon, N)| \le \frac{\varepsilon + 4/\varepsilon}{N}$. Note that for fixed $\varepsilon > 0$, $N = N(h)$ approaches $\infty$ as $h$ approaches 0. Justify the following statement: If $\displaystyle\lim_{h\rightarrow 0} \frac{\mathcal{F}(x+h) - \mathcal{F}(x)}{h}$ exists, then
\begin{equation}\label{2.16}
\lim_{N\rightarrow \infty} \frac{\cos(N! x + \varepsilon) - \cos(N! x)}{\varepsilon}
\end{equation}
is independent of $\varepsilon$.
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 2.2.9})\\
Show that
\begin{equation}\label{2.17}
\frac{\cos(N! x + 2\varepsilon)-\cos(N! x)}{2\varepsilon} - \frac{\cos(N! x + \varepsilon) -\cos(N! x)}{\varepsilon} = \frac{\cos(\varepsilon)-1}{\varepsilon} \cos(N! x + \varepsilon).\end{equation}
Therefore, the limit in \eqref{2.16} is independent of $\varepsilon$ if and only if
$$\lim_{N\rightarrow\infty} \cos(N! x+\varepsilon) = 0$$
regardless of the value of $\varepsilon > 0$, and this is not true for any $x$.
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 2.3.1})\\
Find the derived set of each of the following sets. Which of these sets are first species?
\begin{enumerate}
\item $\mathbb{Q}\cap [0,1]$
\item $[0,1] -\mathbb{Q}$
\item $\{\frac{n}{n+1}\, \mid\, n\in\mathbb{N}\}$
\item $\{\frac{2k-1}{2^n}\,\mid n\in\mathbb{N}, 1\le k\le 2^{n-1}\}$
\item $\{\frac{k}{n}\,|\,n\in\mathbb{N}, k = 1, 2, \mbox{ or } 3\}$
\item $(0,1)\cup(3,4)$
\item $\displaystyle\bigcup_{n=1}^{\infty} \left(\frac{1}{2n},\frac{1}{2n-1}\right) $
\end{enumerate}

\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 2.3.2})\\
Find the outer content of each of the sets in Exercise 2.3.1. Justify each answer.
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 2.3.5})\\
Find the oscillation at $x=1/3$ of the function
\[
g(x) = \begin{cases}
x, & x\in\mathbb{Q},\\
-x, & x\not\in\mathbb{Q}.\end{cases}
\]
Justify your answer.
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 2.3.10})\\
Prove that for any set $S$, $$S''\subseteq S',$$ the derived set of the derived set of $S$ is contained in the derived set of $S$.
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 2.3.11})\\
Prove that the derived set of $T = \{\frac{1}{m} + \frac{1}{n}\,|\, m,n\in\mathbb{N}\}$ is the set $U = \{\frac{1}{n}\,|\,n\in\mathbb{N}\}\cup\{0\}$. First show that $U\subseteq T'$. To prove that $T$ has no other limit points, show that there are only finitely many points of $T$ in $(1/N, 1/(N-1))$ that are not of the form $1/N + 1/n$.
\end{problem}
\begin{sol}
\end{sol}


\end{document} 