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\begin{document}

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\noindent{\Large{\textsc{Intro to Abstract Mathematics\hfill Assignment 9.1  \\ Spring 2020 \hfill Due April 8}}}

\large

\vspace{1cm}


\begin{exercise}
Consider the sequence $a_0, a_1, a_2, \ldots$ of real numbers defined by
\[
\begin{aligned}
a_0 &= 0,\\
a_{n+1} &= a_n^2 + \frac{1}{4} \text{ for all }n \in \N.
\end{aligned}
\]
\begin{itemize}

\item[\bf a.]  Compute $a_1$, $a_2$ and $a_3$.

\item[\bf b.]  Use induction to prove that $0 < a_n < 1/2$ for all $n \ge 1$.

\item[\bf c.]  Prove that $a_{n+1} > a_n$ for all $n \in \N$.  [{\em Suggestion:}  First prove that $x^2 + 1/4 > x$ for all real $x \ne 1/2$.]

\end{itemize}

\end{exercise}

\begin{exercise}
Prove that for all $n \ge 1$, $\displaystyle \sum_{k=1}^n \frac{1}{k^2} \le 2 - \frac{1}{n}$.
\end{exercise}



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