3.9.E: Problems on Boundedness and Diameters (Exercises)
( \newcommand{\kernel}{\mathrm{null}\,}\)
Show that if a set A in a metric space is bounded, so is each subset B⊆A.
Prove that if the sets A1,A2,…,An in (S,ρ) are bounded, so is
n⋃k=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.
⇒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,
n⋃k=1G(pk;εk).
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.]
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.]
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,¯y∈Gp(ε).
To show that there is no smaller upper bound, prove that any number
2ε−2r(r>0)
is exceeded by some ρ(¯x,¯y); e.g., take ¯x and ¯y on some line through ¯p,
¯x=¯p+t→u,
choosing suitable values for t to get ρ(¯x,¯y)=|¯x−¯y|>2ε−2r.]
Prove that in En, a set A is bounded iff it is contained in an interval.
Prove that for all sets A and B in (S,ρ) and each p∈S
ρ(A,B)≤ρ(A,p)+ρ(p,B).
Disprove
ρ(A,B)<ρ(A,p)+ρ(p,B)
by an example.
Find supxn,infxn,maxxn, and minxn (if any) for sequences with general term
(a) n;
(b) (−1)n(2−22−n);
(c) 1−2n;
(d) n(n−1)(n+2)2.
Which are bounded in E1?
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).
Let f:E1→E1 be given by
f(x)=1x if x≠0, and f(0)=0.
Show that f is bounded on an interval [a,b] iff 0∉[a,b]. Is f bounded on (0,1)?
Prove the following:
(a) If A⊆B⊆(S,ρ), then dA≤dB.
(b) dA=0 iff A contains at most one point.
(c) If A∩B≠∅, then
d(A∪B)≤dA+dB.
Show by an example that this may fail if A∩B=∅.