% LaTeX file -- AM/PM 331, Fall 2011, Assignment 1

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%%%%%%%   TITLE   %%%%%%
\begin{center}
\textbf{\large AMath/PMath 331 \\ Assignment 1\\
Due Friday September 23}
\end{center}
\vskip 2em 

\begin{Ex}

\item Let $a_n = \sqrt{n^2-n} - n$ for $n\ge1$.
\begin{parts}
\item Compute $L := \dlim_{n\to\infty} a_n$.
\item Estimate the error $|L-a_n|$, and find an integer $N$ so that
\[ |L-a_n| < \tfrac12 10^{-6} \qforal n\ge N. \]
\end{parts}

\item Let $a_n = \sin(\log n)$ for $n \ge 1$.
\begin{parts}
\item Show that there is an integer in the interval $[e^{\pi(k+\frac13)}, e^{\pi(k+\frac23)}]$
for each $k\ge0$. 
\item What does part (a) say about certain terms of this sequence?
\item Prove that the sequence $(a_n)$ does not converge.
\end{parts}


\item Let $S = \{ x : 0 < \sin(1/x) < 1/2 \}$.  Find $\inf S$.


\item Let $a_1=0$ and $a_{n+1} = \sqrt{5+2a_n}$ for $n \ge 1$.
\begin{parts}
\item Show by induction that $a_n < a_{n+1} < 4$.
\item Prove that $L := \dlim_{n\to\infty} a_n$ exists, and calculate it.
\item Show that $|L-a_{n+1}| < \frac12 |L-a_n|$.
Hence find an integer $N$ so that 
\[ |L-a_n| < \tfrac12 10^{-12} \qforal n \ge N .\]
\end{parts}


%\item Let $f(x) = e^{-x^2}$. Set $a_0=0$ and $a_n = f(a_{n-1})$ for $n \ge 1$.
%\begin{parts}
%\item Use calculus to show that there is a unique point $x_0$ such that $f(x_0)=x_0$.
%\item Show that $|f'(x)| < .2$ for all $x\in\bR$.
%\item Use the mean value theorem to show that
%\[ |f(x_0) - f(a_n)| \le \tfrac15 |x_0 - a_n| .\]
%\item Hence show that $\dlim_{n\to\infty} a_n = x_0$, and find an integer $N$ so that
%\[ |x_0 - a_n| < \tfrac12 10^{-20} \qforal n \ge N .\]
%\end{parts}

%\item Suppose that $(a_n)$ is a sequence in $\bR$, and $(L_i)$
%is a sequence so that $\dlim_{i\to\infty} L_i = L$ exists.
%Suppose that for each $i\ge1$, there is a subsequence $(a_{n_{i,j}})_{j=1}^\infty$
%such that $\dlim_{j\to\infty} a_{n_{i,j}} = L_i$.
%Prove that there is a subsequence of $(a_n)$ converging to $L$.\\[1ex]
%\hint For each $k\ge1$, find an $a_{n_k}$ within $1/k$ of $L$ with $n_{k+1}>n_k$
%by first choosing an $L_i$ sufficiently close to $L$ and then choosing a large enough $n_{i,j}$
%with $a_{n_{i,j}}$ sufficiently close to $L_i$.  Check your estimates carefully.

\item Let $S$ be a non-empty subset of $\bR$ which is bounded above.
Let $L = \sup S$.  
\begin{parts}
\item Prove that there is a sequence $(s_n)$ consisting of
points in $S$ such that $\dlim_{n\to\infty} s_n = L$.
\item Is it always possible to choose the $s_n$ to be distinct points in $S$?
\end{parts}


\end{Ex}
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