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\begin{center}\textbf{Problem Set 8 \hspace{.7in} Real Analysis 1 \hspace{.7in} Due 11/28/18} \end{center}

\noindent Four of these problems will be graded, with each problem worth 5 points.  Clear and complete justification is required for full credit.  You are welcome to discuss these problems with anyone and everyone, but must write up your own final submission without reference to any sources other than the textbook and instructor. 

\begin{enumerate}

\item \, [6.1.3] Prove that a constant function $f(x) = c, c \in \mathbb{R}$, is Riemann integrable on any interval $[a,b]$ and $\int_a^b f(x) \, dx = c(b-a).$
\item \, [6.1.4] If $f(x) \leq g(x) \leq h(x)$ for all $x \in [a,b]$, and $f$ and $h$ are Riemann integrable on $[a,b]$, then so is $g$.
\item \, [6.1.6] If a function $f:[a,b] \to \mathbb{R}$ is bounded and nonnegative, then $\underline{\int_a^b} f \geq 0$.
\item If $f$ and $g$ are Riemann integrable on $[a,b]$ then so is $f-g$, and $\int_a^b (f-g) = \int_a^b f - \int_a^b g.$
\item If $f$ and $g$ are Riemann integrable on $[a,b]$ and $f(x) \le g(x)$ for all $x \in [a,b]$, then $\int_a^b f \le \int_a^b g$.
\item Is the converse of \#5 true?


\end{enumerate}

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