
# 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 $$A \cap B=\emptyset$$.