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\begin{center}
{
Math 2106-D, Foundations of Mathematical Proof \\
Homework 4
\\ Due October 5, 2017
}
\end{center}

\noindent
{\bf 
Do the following problems from Hammack:
}

Chapter 10: 4,16,20,26

Section 11.0: 10

Section 11.1: 4,8

Section 11.2: 12

Section 11.3: 4


\noindent{\bf Also turn in the following exercises:} 

\begin{enumerate}[label=A\arabic*]

\item Suppose that a football team can only score field goals (worth $3$ points) and touchdowns (worth $7$ points). Use induction to show that the football team can score any integral number of points $n\geq12$ (of course, this isn't practical in a real game, but suppose that a game can go on indefinitely). (Hint: Show that it is possible to score $12$, $13$, or $14$ points, and start the inductive hypothesis after this point.)

\item Recall that the $n${\it-th harmonic number} is the finite sum 
\[
H_n=
1+\frac12+\frac13+\ldots+\frac{1}{n-1}+\frac{1}{n}=\sum_{j=1}^n\frac1j
.
\]
Use induction to show that $H_{2^n}\geq1+n/2$. Conclude that the harmonic series $1+1/2+1/3+\ldots$ diverges. (Hint: write down the first few cases of the inequality you need for the inductive step with explicit numbers). 

\end{enumerate}
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