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\section{Q4}  \subsection{a}  \subsubsection{Lemma \subsubsection*{Lemma  1: $\tau \in (0,2\pi/7)$} \textbf{Pf}: From problem definition/assumed finite k we have $\tau > 0$. Additionally: 
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$\square$  \subsubsection{Lemma \subsubsection*{Lemma  2: $\frac{1}{\cos x - \sin x}$ is monotone decreasing for $x\in (0, 2\pi/7)$.} \textbf{Pf}: $\frac{d(\cos x - \sin x)}{dx}=-(\sin x + \cos x) < 0$ for $x\in (0, \pi/4)$ therefore $\cos x - \sin x$ is monotone decreasing on this domain. As such, $\frac{1}{\cos x - \sin x}$ is monotone increasing $ \square$.  \subsubsection{Main \subsubsection*{Main  Proof}