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Mathematics LibreTexts

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

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Exercise 3.9.E.1

Show that if a set A in a metric space is bounded, so is each subset BA.

Exercise 3.9.E.2

Prove that if the sets A1,A2,,An in (S,ρ) are bounded, so is
nk=1Ak.
Disprove this for infinite unions by a counterexample.
[Hint: By Theorem 1, each Ak is in some Gp(εk), with one and the same center p. If the number of the globes is finite, we can put max (ε1,,εn)=ε, so Gp(ε) contains all Ak. Verify this in detail.

Exercise 3.9.E.3

3. From Problems 1 and 2 show that a set A in (S,ρ) is bounded iff it is contained in a finite union of globes,
nk=1G(pk;εk).

Exercise 3.9.E.4

A set A in (S,ρ) is said to be totally bounded iff for every ε>0 (no matter how small), A is contained in a finite union of globes of radius ε. By Problem 3, any such set is bounded. Disprove the converse by a counterexample.
[Hint: Take an infinite set in a discrete space.]

Exercise 3.9.E.5

Show that distances between points of a globe ¯Gp(ε) never exceed 2ε. (Use the triangle inequality!) Hence infer that dGp(ε)2ε. Give an example where dGp(ε)<2ε. Thus the diameter of a globe may be less than twice its radius.
[Hint: Take a globe Gp(12) in a discrete space.]

Exercise 3.9.E.6

Show that in En( as well as in Cn and any other normed linear space  {0} ), the diameter of a globe Gp(ε) always equals 2ε (twice its radius).
[Hint: By Problem 5,2ε is an upper bound of all ρ(¯x,¯y) with ¯x,¯yGp(ε).
To show that there is no smaller upper bound, prove that any number
2ε2r(r>0)
is exceeded by some ρ(¯x,¯y); e.g., take ¯x and ¯y on some line through ¯p,
¯x=¯p+tu,
choosing suitable values for t to get ρ(¯x,¯y)=|¯x¯y|>2ε2r.]

Exercise 3.9.E.7

Prove that in En, a set A is bounded iff it is contained in an interval.

Exercise 3.9.E.8

Prove that for all sets A and B in (S,ρ) and each pS
ρ(A,B)ρ(A,p)+ρ(p,B).
Disprove
ρ(A,B)<ρ(A,p)+ρ(p,B)
by an example.

Exercise 3.9.E.9

Find supxn,infxn,maxxn, and minxn (if any) for sequences with general term
(a) n;
(b) (1)n(222n);
(c) 12n;
(d) n(n1)(n+2)2.
Which are bounded in E1?

Exercise 3.9.E.10

Prove the following about lines and line segments.
(i) Show that any line segment in En is a bounded set, but the entire line is not.
(ii) Prove that the diameter of L(¯a,¯b) and of (¯a,¯b) equals ρ(¯a,¯b).

Exercise 3.9.E.11

Let f:E1E1 be given by
f(x)=1x if x0, and f(0)=0.
Show that f is bounded on an interval [a,b] iff 0[a,b]. Is f bounded on (0,1)?

Exercise 3.9.E.12

Prove the following:
(a) If AB(S,ρ), then dAdB.
(b) dA=0 iff A contains at most one point.
(c) If AB, then
d(AB)dA+dB.
Show by an example that this may fail if AB=.


3.9.E: Problems on Boundedness and Diameters (Exercises) is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by LibreTexts.

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