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3.10.E: Problems on Cluster Points and Convergence (Exercises)

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Exercise $$\PageIndex{1}$$

Is the Archimedean property (see Chapter 2, §10) involved in the proof that
$\lim _{m \rightarrow \infty} \frac{1}{m}=0 ?$

Exercise $$\PageIndex{2}$$

Prove Note 2 and Corollaries 4 and 6.

Exercise $$\PageIndex{3}$$

Verify Example (c) in detail.

Exercise $$\PageIndex{4}$$

Prove Corollary $$5 .$$
[Hint: Fix some $$G_{p}(\varepsilon) .$$ Use Definition 2. If $$G_{p}(\varepsilon)$$ leaves out $$x_{1}, x_{2}, \ldots, x_{k},$$ take a larger radius $$r$$ greater than
$\rho\left(x_{m}, p\right), \quad m=1,2, \dots, k.$
Then the enlarged globe $$G_{p}(r)$$ contains all $$x_{m} .$$ Use Theorem 1 in §13.]

Exercise $$\PageIndex{5}$$

Show that $$x_{m}=m$$ tends $$\mathrm{to}+\infty$$ in $$E^{*} .$$ Does it contradict Corollary 5$$?$$

Exercise $$\PageIndex{6}$$

Show that $$E^{1}$$ is a perfect set $$i n E^{1} : E^{1}=\left(E^{1}\right)^{\prime} .$$ Is $$E^{1}$$ a perfect set in $$E^{*} ?$$ Why?

Exercise $$\PageIndex{7}$$

$$\Rightarrow 7 .$$ Review Problems 2 and 4 of Chapter 2, §13. (Do them if not done before.)

Exercise $$\PageIndex{8}$$

Verify Examples (f) and (h).

Exercise $$\PageIndex{9}$$

Explain Example (i) in detail.

Exercise $$\PageIndex{10}$$

In the following cases find the set $$A^{\prime}$$ of all cluster points of $$A$$ in $$E^{1} .$$ Is $$A^{\prime} \subseteq A ?$$ Is $$A \subseteq A^{\prime} ?$$ Is $$A$$ perfect? Give a precise proof.
(a) $$A$$ consists of all points of the form
$\frac{1}{n} \text { and } 1+\frac{1}{n}, \quad n=1,2, \ldots;$
i.e., $$A$$ is the sequence
$\left\{1,2, \frac{1}{2}, 1 \frac{1}{2}, \ldots, \frac{1}{n}, 1+\frac{1}{n}, \ldots\right\}.$
(b) $$A$$ is the set of all rationals in $$(0,1) .$$ Answer: $$A^{\prime}=[0,1] .$$ Why?
(c) $$A$$ is the union of the intervals
$\left[\frac{2 n}{2 n+1}, \frac{2 n+1}{2 n+2}\right], \quad n=0,1,2, \ldots$
(d) $$A$$ consists of all points of the form
$2^{-n} \text { and } 2^{-n}+2^{-n-k}, \quad n, k \in N.$

Exercise $$\PageIndex{11}$$

Can a sequence $$\left\{x_{m}\right\} \subseteq E^{1}$$ cluster at each $$p \in E^{1} ?$$
[Hint: See Example (e). $$]$$

Exercise $$\PageIndex{12}$$

Prove that if
$p=\sup A \text { or } p=\inf A \text { in } E^{1}$
$$\left(\emptyset \neq A \subseteq E^{1}\right),$$ and if $$p \notin A,$$ then $$p$$ is a cluster point of $$A .$$
[Hint: Take $$G_{p}(\varepsilon)=(p-\varepsilon, p+\varepsilon) .$$ Use Theorem 2 of Chapter 2, §§8-9.]

Exercise $$\PageIndex{13}$$

Prove that a set $$A \subseteq(S, \rho)$$ clusters at $$p$$ iff every neighborhood of $$p$$ (see §12, Definition 1) contains infinitely many points of $$A ;$$ similarly for sequences. How about convergence? State it in terms of cubic neighborhoods in $$E^{n} .$$

Exercise $$\PageIndex{14}$$

Discuss Example $$(\mathrm{h})$$ for nondegenerate intervals in $$E^{n} .$$ Give a proof.

Exercise $$\PageIndex{15}$$

Prove that a set $$A \neq \emptyset$$ clusters at $$p(p \notin A)$$ iff $$\rho(p, A)=0 .$$ (See §13, Note $$3 . )$$

Exercise $$\PageIndex{16}$$

Show that in $$E^{n}(* \text { and in any other normed space } \neq\{\overline{0}\}),$$ the cluster points of any globe $$G_{\overline{p}}(\varepsilon)$$ form exactly the closed globe $$\overline{G}_{\overline{p}}(\varepsilon),$$ and that $$\overline{G}_{\overline{p}}(\varepsilon)$$ is perfect. Is this true in other spaces? (Consider a discrete space!)

Exercise $$\PageIndex{17}$$

(Cantor's set.) Remove from $$[0,1]$$ the open middle third
$\left(\frac{1}{3}, \frac{2}{3}\right).$
From the remaining closed intervals
$\left[0, \frac{1}{3}\right] \text { and }\left[\frac{2}{3}, 1\right],$
remove their open middles,
$\left(\frac{1}{9}, \frac{2}{9}\right) \text { and }\left(\frac{7}{9}, \frac{8}{9}\right).$
Do the same with the remaining four closed intervals, and so on, ad infinitum. The set $$P$$ which remains after all these (infinitely many) removals is called Cantor's set.
Show that $$P$$ is perfect.
[Hint: If $$p \notin P,$$ then either $$p$$ is in one of the removed open intervals, or $$p \notin[0,1]$$. In both cases, $$p$$ is no cluster point of $$P$$. (Why?) Thus no $$p$$ outside $$P$$ is a cluster point.
On the other hand, if $$p \in P,$$ show that any $$G_{p}(\varepsilon)$$ contains infinitely many endpoints of removed open intervals, all in $$P ;$$ thus $$p \in P^{\prime} .$$ Deduce that $$P=P^{\prime}$$]