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\flushleft\underline{Homework 9 --- MATH 4590 Spring 2018} \\
\begin{enumerate}[1.]
\item Use induction to prove that
$$1+r+r^2+\ldots+r^n = \dfrac{1-r^{n+1}}{1-r}.$$
\item Define 
$$F(x) = \left\{ \begin{array}{ll}
0, & \quad x<0 \\
(n-1)x - \dfrac{(n-1)n}{2}, & \quad x \in [n-1,n), n \in \{1,2,3,\ldots\}
\end{array} \right.$$
\begin{enumerate}[a.)]
\item Sketch this function on $[0,5]$. Is $F$ continuous?
\item Find $F'(x)$ at places which have a derivative. Add a sketch for $F'$ to your sketch in part a.).
\item Use the above part to evaluate $\displaystyle\int_a^b \lfloor x \rfloor \mathrm{d}x$, where $\lfloor x \rfloor$ denotes the floor function (i.e. $\lfloor x \rfloor$ is the greatest integer $\leq x$)
\end{enumerate}
\item Prove that if $Z_1$ and $Z_2$ are zero sets, then $Z_1 \cup Z_2$ is a zero set.
\item Show that any two antiderivatives of a function $f$ differ by a constant. (\textit{hint: use the ``antiderivative theorem"})
\item If $f_n \rightrightarrows f$ and $g_n \rightrightarrows g$ on $A$, prove that $f_n+g_n \rightrightarrows f+g$ on $A$.
\end{enumerate}
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