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Since $\N$ is countably infinite, Each set is definitely finite, because they all contain just one element. Finally, the union of the collection of sets is equal to $\N$, which is not a finite set. -\exercise* +\exercise*[6] \begin{tcolorbox} \begin{enumerate}[label=\emph{\alph*)}, wide] \item Compute $f(4/15)$. 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zx&WJHtDt2gu1_x(lgygf0NYT>%5cdnl`9RjKMROP^^9{iR$3_bC-rRI!ny$8=0p 0 \; \text{and} \; x^3 < 2\}. + \end{equation*} + + \begin{enumerate}[label=\emph{\alph*)}] + \item Prove that $E$ is bounded above. + \item Let $r = \sup E$ (which exists by part a)). Prove that $r > 0$ and $r^3=2$. + + \textit{Hint:} Adapt the proof used in Example 1.2.3. + \end{enumerate} +\end{tcolorbox} + +So, let $E$ and $r$ be defined as in the exercise statement. Then: +\begin{enumerate}[label=\emph{\alph*)}] + \item $x \leq 2$ is an upper bound for $E$, as $2 * 2 * 2 = 8$. So, $E$ is bounded above. + \item As $1 \in E$, $r \geq 1 > 0$, so the first part of the statement holds. In order to show that $r^3=2$, + we want to show that $r^3 \leq 2$ and $r^3 \geq 2$ hold. + + First, let's show that $r^3 \geq 2$. We will take a similar approach as in Example 1.2.3 from the textbook. + So, take a positive number $s$ such that $s^3 < 2$. We wish to find an $h > 0$ such that $(s + h)^3 < 2$. + As $2 - s^3 > 0$, we have $\frac{2 - s^3}{3s^2 + 3s + 1} > 0$. Choose an $h \in \R$ such that $0 < h < + \frac{2 - s^3}{3s^2+3s+1}$. Furthermore, assume $h < 1$. Estimate, + + \begin{align*} + (s+h)^3 - s^3 &= h(3s^2 + 3sh + h^2) \\ + &< h(3s^2 + 3s + 1) \quad &\text{(since } h < 1) \\ + &< 2 - s^3 \quad &\text{(since } h < \frac{2 - s^3}{3s^2 + 3s + 1}). + \end{align*} + + Therefore, $(s+h)^3 < 2$. Hence $s + h \in E$, but as $h > 0$, we have $s + h > s$. So $s < r = \sup E$. + As $s$ was an arbitrary positive number such that $s^3 < 2$, it follows that $r^3 \geq 2$. + + Now take an arbitrary positive number $s$ such that $s^3 > 2$. We wish to find an $h > 0$ such that + $(s-h)^3 > 2$ and $s - h$ is still positive. As $s^3 - 2 > 0$, we have that $\frac{s^3 - 2}{3s^2+1} > 0$. + Let $h := \frac{s^3-2}{3s^2+1}$, and check that $s - h = s - \frac{s^3-2}{3s^2+1} = + \frac{2s^3+s+2}{3s^2+1} > 0$. Assume that $h < 1$. Estimate, + + \begin{align*} + s^3 - (s-h)^3 &= h(3s^2 - 3sh + h^2) \\ + &< h(3s^2+h^2) \quad &\text{(since } s>0 \; \text{ and } h > 0) \\ + &< h(3s^2+1) \quad &\text{(since } h < 1) \\ + &= s^3 - 2 &\text{(because of the definition of } h). + \end{align*} + + By subtracting $s^3$ from both sides and multiplying by -1, we find $(s-h)^3>2$. Therefore, $s-h \notin E$. + Moreover, if $x \geq s-h$, then $x^3 \geq (s-h)^3 > 2$ (as $x > 0$ and $s-h > 0$) and so $x \notin E$. Thus, + $s - h$ is an upper bound for $E$. However, $s-h < s$, or in other words, $s > r = \sup E$. + Hence, $r^3 \leq 2$. + + Together, $r^3 \geq 2$ and $r^3 \leq 2$ imply $r^3 = 3$. \qed +\end{enumerate} \end{document}

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