Section 2.8: Problem 5 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James $^{*}\mathfrak{R}$ . Munkres

(Bolzano–Weierstrass theorem) Let $A\subseteq\mathbb{R}$ be bounded and infinite. Show that there is a point $p\in\mathbb{R}$ that is infinitely close to, but different from, some member of $^{∗}A$ . Suggestion: Let $S:\mathbb{N}\rightarrow A$ with $S$ one-to-one; look at $^{∗}S(x)$ for infinite $x\in^{∗}\mathbb{N}$ .

As suggested, consider a one-to-one function $S:\mathbb{N}\rightarrow A$ (there exists one as $A$ is infinite). Then, according to Exercise 2(a), $^{*}S:{}^{*}\mathbb{N}\rightarrow^{*}\mathbb{R}$ . In fact, $^{*}S:{}^{*}\mathbb{N}\rightarrow^{*}A$ , as the sentence $\forall v_{1}\forall v_{2}(P_{S}v_{1}v_{2}\rightarrow P_{A}v_{2})$ is true in $\mathfrak{R}$ , and, hence, in $^{*}\mathfrak{R}$ . Moreover, $^{*}A$ is bounded, as for some $r,s\in\mathbb{R}$ , the sentence $\forall v_{1}(P_{A}v_{1}\rightarrow c_{r}<v_{1}<c_{s})$ is true in $\mathfrak{R}$ , and, hence, in $^{*}\mathfrak{R}$ . Suppose $x\in^{∗}\mathbb{N}$ is infinite, and let $b={}^{*}S(x)$ . Then, $r<b<s$ , i.e. $b\in\mathcal{F}$ , and, according to Exercise 3, $b\in{}^{*}A-\mathbb{R}$ , i.e. $p=\mbox{st}b\neq b$ . Overall, we have $p\in\mathbb{R}$ , $b\in{}^{*}A$ , $p\neq b$ and $p\simeq b$ .

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Section 2.8: Problem 5 Solution

Section 2.8: Problem 5 Solution