3.9.E: Problems on Boundedness and Diameters (Exercises)

Exercise $\PageIndex{1}$

Show that if a set $A$ in a metric space is bounded, so is each subset $B \subseteq A .$

Exercise $\PageIndex{2}$

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.

Exercise $\PageIndex{3}$

$\Rightarrow 3 .$ 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,
$$\bigcup_{k=1}^{n} G\left(p_{k} ; \varepsilon_{k}\right).$$

Exercise $\PageIndex{4}$

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.]

Exercise $\PageIndex{5}$

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.]

Exercise $\PageIndex{6}$

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 $\rho(\overline{x}, \overline{y})=|\overline{x}-\overline{y}|>2 \varepsilon-2 r . ]$

Exercise $\PageIndex{7}$

Prove that in $E^{n},$ a set $A$ is bounded iff it is contained in an interval.

Exercise $\PageIndex{8}$

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.

Exercise $\PageIndex{9}$

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 $E^{1} ?$

Exercise $\PageIndex{10}$

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 $\rho(\overline{a}, \overline{b})$.

Exercise $\PageIndex{11}$

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 $(0,1) ?$

Exercise $\PageIndex{12}$

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 $A \cap B=\emptyset$.