3.10.E: Problems on Cluster Points and Convergence (Exercises)
Is the Archimedean property (see Chapter 2, §10) involved in the proof that
\[
\lim _{m \rightarrow \infty} \frac{1}{m}=0 ?
\]
Prove Note 2 and Corollaries 4 and 6.
Verify Example (c) in detail.
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.]
Show that \(x_{m}=m\) tends \(\mathrm{to}+\infty\) in \(E^{*} .\) Does it contradict Corollary 5\(?\)
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?
\(\Rightarrow 7 .\) Review Problems 2 and 4 of Chapter 2, §13. (Do them if not done before.)
Verify Examples (f) and (h).
Explain Example (i) in detail.
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
\[
2^{-n} \text { and } 2^{-n}+2^{-n-k}, \quad n, k \in N.
\]
Can a sequence \(\left\{x_{m}\right\} \subseteq E^{1}\) cluster at each \(p \in E^{1} ?\)
[Hint: See Example (e). \(]\)
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.]
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 \(E^{n} .\)
Discuss Example \((\mathrm{h})\) for nondegenerate intervals in \(E^{n} .\) Give a proof.
Prove that a set \(A \neq \emptyset\) clusters at \(p(p \notin A)\) iff \(\rho(p, A)=0 .\) (See §13, Note \(3 . )\)
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!)
(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 \(P=P^{\prime}\)]