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

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 $P=P^{\prime}$]