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Chapter 2: Problem 15
Find the decimal representation of \(-\frac{2}{7}\).
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Let \(S \subseteq \mathbb{R}\) be nonempty. Show that \(u \in \mathbb{R}\) is an upper bound of \(S\) if and only if the conditions \(t \in \mathbb{R}\) and \(t>u\) imply that \(t \notin S\).
If \(0 \leq a
Let \(S \subseteq \mathbb{R}\) and suppose that \(s^{*}:=\sup S\) belongs to \(S\). If \(u \notin S\), show that \(\sup (S \cup\\{u\\})=\) \(\sup \left\\{s^{*}, u\right\\} .\)
If \(I_{1} \supseteq I_{2} \supseteq \cdots \supseteq I_{n} \supseteq \cdots\) is a nested sequence of intervals and if \(I_{n}=\left[a_{n}, b_{n}\right]\), show that \(a_{1} \leq a_{2} \leq \cdots \leq a_{n} \leq \cdots\) and \(b_{1} \geq b_{2} \geq \cdots \geq b_{n} \geq \cdots\)
If \(a, b \in \mathbb{R}\), prove the following. \(\Rightarrow 1,1-2\) (a) If \(a+b=0\), then \(b=-a, \quad 4\) b, 4 (b) \(-(-a)=a\) (c) \((-1) a=-a\), (d) \((-1)(-1)=1 .\)
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