3.9.E: Problems on Boundedness and Diameters (Exercises)
Show that if a set \(A\) in a metric space is bounded, so is each subset \(B \subseteq A .\)
Prove that if the sets \(A_{1}, A_{2}, \ldots, A_{n}\) in \((S, \rho)\) are bounded, so is
\[
\bigcup_{k=1}^{n} A_{k}.
\]
Disprove this for infinite unions by a counterexample.
[Hint: By Theorem \(1,\) each \(A_{k}\) is in some \(G_{p}\left(\varepsilon_{k}\right),\) with one and the same center \(p\). If the number of the globes is finite, we can put max \(\left(\varepsilon_{1}, \ldots, \varepsilon_{n}\right)=\varepsilon,\) so \(G_{p}(\varepsilon)\) contains all \(A_{k} .\) Verify this in detail.
\(\Rightarrow 3 .\) From Problems 1 and 2 show that a set \(A\) in \((S, \rho)\) is bounded iff it is contained in a finite union of globes,
\[
\bigcup_{k=1}^{n} G\left(p_{k} ; \varepsilon_{k}\right).
\]
A set \(A\) in \((S, \rho)\) is said to be totally bounded iff for every \(\varepsilon>0\) (no matter how small), \(A\) is contained in a finite union of globes of radius \(\varepsilon\). By Problem 3, any such set is bounded. Disprove the converse by a counterexample.
[Hint: Take an infinite set in a discrete space.]
Show that distances between points of a globe \(\overline{G}_{p}(\varepsilon)\) never exceed 2\(\varepsilon\). (Use the triangle inequality!) Hence infer that \(d G_{p}(\varepsilon) \leq 2 \varepsilon\). Give an example where \(d G_{p}(\varepsilon)<2 \varepsilon .\) Thus the diameter of a globe may be less than twice its radius.
[Hint: Take a globe \(G_{p}\left(\frac{1}{2}\right)\) in a discrete space.]
Show that in \(E^{n}\left(* \text { as well as in } C^{n} \text { and any other normed linear space }\right.\) \(\neq\{0\}\) ), the diameter of a globe \(G_{p}(\varepsilon)\) always equals 2\(\varepsilon\) (twice its radius).
[Hint: By Problem \(5,2 \varepsilon\) is an upper bound of all \(\rho(\overline{x}, \overline{y})\) with \(\overline{x}, \overline{y} \in G_{p}(\varepsilon) .\)
To show that there is no smaller upper bound, prove that any number
\[
2 \varepsilon-2 r \quad(r>0)
\]
is exceeded by some \(\rho(\overline{x}, \overline{y}) ;\) e.g., take \(\overline{x}\) and \(\overline{y}\) on some line through \(\overline{p}\),
\[
\overline{\boldsymbol{x}}=\overline{\boldsymbol{p}}+\boldsymbol{t} \overrightarrow{\boldsymbol{u}},
\]
choosing suitable values for \(t\) to get \(\rho(\overline{x}, \overline{y})=|\overline{x}-\overline{y}|>2 \varepsilon-2 r . ]\)
Prove that in \(E^{n},\) a set \(A\) is bounded iff it is contained in an interval.
Prove that for all sets \(A\) and \(B\) in \((S, \rho)\) and each \(p \in S\)
\[
\rho(A, B) \leq \rho(A, p)+\rho(p, B).
\]
Disprove
\[
\rho(A, B)<\rho(A, p)+\rho(p, B)
\]
by an example.
Find \(\sup x_{n}, \inf x_{n}, \max x_{n},\) and \(\min x_{n}\) (if any) for sequences with general term
(a) \(n\);
(b) \((-1)^{n}\left(2-2^{2-n}\right)\);
(c) \(1-\frac{2}{n}\);
(d) \(\frac{n(n-1)}{(n+2)^{2}}\).
Which are bounded in \(E^{1} ?\)
Prove the following about lines and line segments.
(i) Show that any line segment in \(E^{n}\) is a bounded set, but the entire line is not.
(ii) Prove that the diameter of \(L(\overline{a}, \overline{b})\) and of \((\overline{a}, \overline{b})\) equals \(\rho(\overline{a}, \overline{b})\).
Let \(f : E^{1} \rightarrow E^{1}\) be given by
\[
f(x)=\frac{1}{x} \text { if } x \neq 0, \text { and } f(0)=0.
\]
Show that \(f\) is bounded on an interval \([a, b]\) iff 0\(\notin[a, b] .\) Is \(f\) bounded on \((0,1) ?\)
Prove the following:
(a) If \(A \subseteq B \subseteq(S, \rho),\) then \(d A \leq d B\).
(b) \(d A=0\) iff \(A\) contains at most one point.
(c) If \(A \cap B \neq \emptyset\), then
\[
d(A \cup B) \leq d A+d B.
\]
Show by an example that this may fail if \(A \cap B=\emptyset\).