328 CHAPTER 15. SEQUENCES, COMPACTNESS, CONTINUITY

22. Find limx→0sin(|x|)|x| and prove your answer from the definition of limit.

23. Suppose g is a continuous vector valued function of one variable defined on [0,∞).Prove

limx→x0

g (|x|) = g (|x0|) .

24. Let U = {(x,y,z) such that z > 0}. Determine whether U is open, closed or neither.

25. Let U = {(x,y,z) such that z ≥ 0} . Determine whether U is open, closed or neither.

26. Let U ={(x,y,z) such that

√x2 + y2 + z2 < 1

}. Tell whether U is open, closed or

neither.

27. Let U ={(x,y,z) such that

√x2 + y2 + z2 ≤ 1

}. Tell whether U is open, closed or

neither.

28. Show carefully that Rp is both open and closed.

29. Show that every non empty open set in Rp is the union of open balls contained in it.

30. Show the intersection of any two open sets is an open set.

31. Closed sets were defined to be those sets which are complements of open sets. Showthat a set is closed if and only if it contains all its limit points.

32. Prove the extreme value theorem, a continuous function achieves its maximum andminimum on any closed and bounded set C. Hint: Suppose λ = sup{ f (x) : x ∈C}.Then there exists {xn} ⊆ C such that f (xn)→ λ . Now select a convergent subse-quence. Do the same for the minimum.

33. If C is a collection of open sets such that ∪C ⊇ H a closed and bounded set. ALebesgue number δ is one which has the property that if x ∈ H, then B(x,δ ) iscontained in some set of C . Show that there exists a Lebesgue number. Hint:If thereis no Lebesgue number, then for each n ∈ N, 1/n is not a Lebesgue number. Hencethere exists xn ∈ H such that B(xn,1/n) is not contained in a single set of C . Extracta convergent subsequence, still denoted as xn → x. Then B(x,δ ) is contained in asingle set of C . Isn’t it the case that B(xn,1/n) is contained in B(x,δ ) for all n largeenough? Isn’t this a contradiction?

34. Let C be a closed and bounded set and suppose f : C → Rm is continuous. Showthat f must also be uniformly continuous. This means: For every ε > 0 there existsδ > 0 such that whenever x,y ∈ C and |x−y| < δ , it follows |f (x)−f (y)| < ε .It is in the chapter but go over it again. This is a good time to review the definitionof continuity so you will see the difference. Hint: Suppose it is not so. Then thereexists ε > 0 and {xk} and {yk} such that |xk −yk|< 1

k but |f (xk)−f (yk)| ≥ ε .

35. A set K is compact means that if C is a set of open sets such that ∪C ⊇ K, then thereexists a finite subset {U1, · · · ,Un} ⊆C such that ∪n

i=1Ui ⊇ K. Show every closed andbounded set K in Rp is compact. (Open covers admit finite sub covers.) Next showthat if a set in Rp is compact, then it must be closed and bounded. This is called theHeine Borel theorem. Hint: To show closed and bounded is compact, you might use