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\begin{document}
{\large
\begin{center}
{\bf MAT 137Y: Calculus with proofs}\\
{\bf Assignment 6} \\
{\bf Due on Thursday, January 28 by 11:59pm via Crowdmark}
\end{center}
}
\vv
\begin{quotation}
{\bf Instructions:}
\begin{itemize}
\item You will need to submit your solutions electronically via Crowdmark. \href{https://www.math.toronto.edu/~alfonso/137/PS/137_CM.html}{See MAT137 Crowdmark help page for instructions}. Make sure you understand how to submit and that you try the system ahead of time. If you leave it for the last minute and you run into technical problems, you will be late. There are no extensions for any reason.
\item You may submit individually or as a team of two students. See the link above for more details.
\item You will need to submit your answer to each question separately.
\item This problem set is about Unit 7.
\end{itemize}
\end{quotation}
\
The goal of this assignment is to prove the following result from the definition of integral:
\begin{quotation}
\noindent {\bf Theorem 1:}
Let $a**0$, there exists a partition $P$ of $[a,b]$ such that
$$
\underline{I_a^b}(f) - \e \; < \; L_P(f).
$$
\emph{Note:} This is a very, very short proof if you understand the definition of lower integral as supremum. You may even have learned something similar in class. You do not need to submit your answer to this question, but we want to make sure you think about it before trying the harder (and related) next question.
\item Prove that for every $\e>0$, there exists a partition $P$ of $[a,b]$ such that
$$
\underline{I_a^b}(f) \, + \, \underline{I_a^b}(g) \, - \, \e \; < L_P(f) \, + \, L_P(g).
$$
\emph{Hint:} This will feel a bit like those ``$\e$-$\delta$" proofs you learned in Unit 2.
\item Prove that
$$
\underline{I_a^b}(f) \; + \; \underline{I_a^b}(g) \; \leq \; \underline{I_a^b}(h).
$$
\emph{Note:} If, at this moment, you think you have proven a strict inequality instead of a non-strict inequality, then your argument is probably wrong.
\item This question is irrelevant to the proof of Theorem 1, but it is also interesting. Is it always true that
$$
\underline{I_a^b}(f) \; + \; \underline{I_a^b}(g) \; = \; \underline{I_a^b}(h) ?
$$
Prove it.
\item {\bf [Do not submit]} Repeat the steps from the previous questions (with upper rather than lower sums and integrals) to prove that
$$
\overline{I_a^b}(h) \; \leq \; \overline{I_a^b}(f) \; + \; \overline{I_a^b}(g).
$$
\item Prove Theorem 1.
\end{enumerate}
\end{document}
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