Math 4200: Homework Problems

Gregor Kovacic

1. Prove the following properties of the binomial coefficients

(i) 1 +(n

1

)+

(n

2

)+ · · ·+

(n

n− 1

)+

(n

n

)= 2n,

(ii)(n

1

)+ 2

(n

2

)+ 3

(n

3

)+ · · ·+ n

(n

n

)= n2n−1,

(iii)(n

0

)2

+(n

1

)2

+ · · ·+(n

n

)2

=

(2n

n

)

HINT: In (ii), write the binomial coefficients in terms of factorials. In (iii), consider thecoefficient of xn in (1 + x)2n.

2. For q 6= 1, prove by induction that

1 + 2q + 3q2 + ·+ nqn−1 =1− (n + 1)qn + nqn+1

(1− q)2.

3. If r is rational r 6= 0 and x is irrational, prove that r + x and rx are irrational.

4. Prove that√

3 is irrational.

5. Prove that for any rational root of a polynomial with integer coefficients,

anxn + an−1x

n−1 + · · ·+ a1x + a0, an 6= 0,

if written in lowest terms (i.e., with no common integer factor) as p/q, that the denumeratorp is a factor of a0 and the denominator q is a factor of an.

1

6. If n is a positive integer, show that

an =

(1 +

√5)n −

(1−

√5)n

2n√

5

is a positive integer.

HINT: Show that an+2 − an+1 − an = 0.

7. If a, b > 0 and an > bn, where n is a positive integer, prove that a > b.

8. Prove that there is no real number x such that x2 = −1.

9. Show that for positive a, b, and c,

(i) a2 + b2 + c2 ≥ ab + bc + ca,

(ii) (a + b)(b + c)(c + a) ≥ 8abc.

10. Show that √(a1 − b1)2 + (a2 − b2)2 ≤

√a2

1 + a22 +

√b21 + b2

2,

and interpret this inequality geometrically.

HINT: Show the result for the square of this inequality first, using the Cauchy-Schwartzinequality.

11. Show that even positive integers form a countable set.

12. Show that the set of irrational numbers is uncountable.

13. (i) If z is a complex number such that |z| = 1, compute |1 + z|2 + |1− z|2.

(ii) For complex a and b, show that

|a + b|2 + |a− b|2 = 2(|a|2 + |b|2),

and interpret this result geometrically.

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14. Find all the solutions of the equation z3 = 1. Write the real and imaginary parts ofthese roots in terms of fractions involving integers and square roots of integers.

15. If z = x + iy, x, y real, show that 1/z is also of the form 1/z = u + iv, u, v real.

16. If a and b are real numbers, b 6= 0, show that

√a + ib = ±

√a +

√a2 + b2

2+ i

b

|b|

√−a +

√a2 + b2

2

,

and explain why the expressions under the square root signs are non-negative.

HINT: Assume (x+ iy)2 = a+ ib, and find two equations for x and y. From these equations,deduce (x2 + y2)2 = a2 + b2, and then deduce the expressions for x2 and y2. Finally, becareful about choosing the relative signs of x and y.

17. If n is a positive integer, and

ω = cos2π

n+ i sin

2π

n,

show that1 + ωh + ω2h + · · ·+ ω(n−1)h = 0

for any integer h which is not a multiple of n. What is the geometric interpretation of thisequality? What happens if h is a multiple of n?

18. Show that the empty set is a subset of every set.

19. (i) If A and B are arbitrary sets, and CA and CB are their complements within somelarger set, prove that CX ∪ CY = C(X ∩ Y ).

(ii) State and prove the same result for an arbitrary number, n, of sets A1, . . . , An.

20. Interpret the cartesian product, R2 = R×R, of two sets of real numbers, R, as a plane.In other words, let

R2 = {(x, y) | x, y ∈ R}.

Let A, B, C ⊂ R be the intervals A = {t ∈ R | |t| < 3}, B = {t ∈ R | |t| < 2}, C = {t ∈ R ||t| < 1}.

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(i) Draw the set A×B − C × C.

(ii) Draw the set ((A−B)× ((A−B) ∪ C)) ∪ {(x, y) | x2 + y2 < 1}.

21. Show that, in C, a definition of an open set equivalent to the one given in class is: A ⊂ Cis open if every point z ∈ A is contained in a square S with sides parallel to the real andimaginary axes such that z ∈ S ⊂ A.

HINT: You can inscribe a square into any circle and vice versa.

22. Construct a countable infinity of open sets An ⊂ C, n = 1, 2, 3, . . . , such that theirintersection ∩∞n=1An is not open.

23. Show that the square S = {x + iy | 0 < x < 1, 0 < y < 1} is an open subset of C.

24. Give an example of a subset of the real axis with precisely two cluster points.

25. Is every finite subset of R or C closed? Justify your answer.

26. Are the following subsets of the real axis open, closed, or neither:

(i) A = {x | −2 < x < 3 or 4 < x < 5}.

(ii) B = {x | x(x− 1)(x− 2) ≥ 0}.

(iii) C = {x | 0 < x2 − 1 ≤ 3}.

Justify your answers.

27. Show that a closed ball in R or C is a closed set. What geometric object is such a ballin each case?

28. Draw the subsets A = C ∪D, B = C ∩D of the complex plane and determine whetherthey are open, closed, or neither, where

(i) C = {z | |z − 1| < 2} and D = {z | |z + 1| < 2}.

(ii) C = {z | |z − 1| ≤ 2} and D = {z | |z + 1| ≤ 2}.

(iii) C = {z | |z − 1| < 2} and D = {z | |z + 1| ≤ 2}.

4

Justify your answers.

29. (i) Construct a closed subset of the real axis with precisely three cluster points.

(ii) Show that the set {1/n | n ∈ Z} is neither open nor closed.

(iii) Show that if you remove a finite number of points from an open set, the remaining setis still open. Then show by examples that this may or may not be the case if you remove acountable number of points?

30. Construct a countable collection of open sets in R such that their intersection is neitheropen nor closed.

HINT: You may have to use the fact that a countable union of countable collections is acountable collection.

31. (i) Is every point of every open set E in C a limit point of E?

(ii) Same question for closed sets in C.

(iii) Do the answers of (i) and (ii) change with R in place of C?

32. Let A denote a subset of C.

(i) Do A and its closure A always have the same interior A◦?

(ii) Do A and its interior A◦ always have the same closure A?

Justify your answers. Do these answers change if C is replaced by R?

33. What are the boundary points of Q in R?

HINT: Take it as a fact that any neighborhood of any number contains both rationals andirrationals.

34. Show that the union of two disjoint open or closed subsets of R or C is disconnected.

35. (i) Show that every infinite set of real numbers has a countable dense subset.

(ii) Give an example of a set A ⊂ R such that A ∩Q is not dense in A.

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36. Let E be the set of all x ∈ [0, 1] whose decimal expansion contains only digits 4 and 7.Is E countable? Is E dense in [0, 1]? Is E open? Is E closed? Is E perfect, that is, is everypoint of E a limit point of E? Is E connected? Is E bounded?

37. Find the domains and ranges of the following functions, and determine whether they areone-to-one. If yes, find the inverse functions.

(i) f(x) = x2 + 1.

(ii) f(x) = x3.

(iii) f(x + iy) = xa− yb + i(xb + ya), where a2 + b2 > 0.

(iv) f(x + iy) = ex(cos y + i sin y).

38. Let f : A → B be a function mapping the set A into a set B. If C, D ⊂ A, show thatf(C ∩D) ⊂ f(C) ∩ f(D). Show by example that this inclusion may be proper.

39. Let Xλ = [0, 1] for all λ ∈ [0, 1]. Show that the uncountable Cartesian product∏

λ∈[0,1] Xλ

equals the set of functions f : [0, 1] → [0, 1]. Verify that the axiom of choice holds in thiscase.

40. Let an = 10n/n!.

(i) Is the sequence {an} monotonic? Is it monotonic from a certain n onward?

(i) Show that the sequence converges. To what limit?

(iii) Give an estimate of the difference between an and its limit.

41. Prove that limn→∞

n!

nn= 0.

HINT: Show that 0 <n!

nn<

1

2[n/2], where [x] denotes the largest integer ≤ x.

42. Prove that

(i) limn→∞

(1

n2+

1

(n + 1)2+ · · ·+ 1

(2n)2

)= 0,

(ii) limn→∞

(1√n

+1√

n + 1+ · · ·+ 1√

2n

)= ∞,

6

(iii) if an =1

n+

1

n + 1+ · · ·+ 1

2n, its limit exists and is contained between 1/2 and 1.

HINT: Compare the sums with their largest and smallest terms, respectively, when appro-priate. For (iii), show that {an} is monotonic.

43. Compute limn→∞

(1

1 · 2+

1

2 · 3+ · · ·+ 1

n(n + 1)

).

HINT: Express each term as an appropriate difference.

44. Prove that the sequence√

2,√

2√

2,

√2√

2√

2, . . . converges, and find its limit.

45. Prove that convergence of {an} implies convergence of {|an|}. Is the converse true? Ifyes, prove it, if no, provide a counterexample.

46. Given a sequence {an}, define its arithmetic means by

sn =a1 + · · ·+ an

n.

(i) Show that, if {an} converges, {sn} must converge to the same limit.

HINT: Write

∆n = sn − an =a1 − an

n+

a2 − an

n+ . . . +

an−1 − an

n.

Use the definition of convergence for {an} and the boundedness of {an} to show that ∆n → 0.

(ii) Construct a sequence {an} which does not converge although limn→∞ sn = 0.

47. Let {an} = a1, a2, a3, . . . and {bn} = b1, b2, b3, . . . be two sequences. Show that the set ofcluster points of the sequence a1, b1, a2, b2, a3, b3, . . . is the union of the sets of cluster pointsof the sequences {an} and {bn}.

HINT: Use subsequences. In particular, also use the fact that if some sequence converges,every one of its subsequences converges to the same limit.

48. Let a1 and b1 be any positive numbers, and let a1 < b1. Let a2 and b2 be defined by theequations

a2 =√

a1b1, b2 =a1 + b1

2.

7

Similarly, let

a3 =√

a2b2, b3 =a2 + b2

2,

and, in general,

an =√

an−1bn−1, bn =an−1 + bn−1

2.

Prove that

(i) a1 < a2 < · · · < an < · · · < bn < · · · < b2 < b1 and deduce that the sequences {an} and{bn} converge.

(ii) Show that {an} and {bn} converge to the same limit.

HINT: You may want to use the fact that limn→∞ xn−1 = limn→∞ xn for every convergentsequence {xn}

49. (i) Show that between any two real numbers there is an irrational number.

HINT: Let the two numbers be a < b. Map the closed interval [a, b] onto [0, 1] by a linearfunction to show that [a, b] is uncountable. What would happen if [a, b] contained nothingbut rationals?

(ii) Show that irrationals are dense in R.

50. Prove that the limit of the sequence√

2,√

2 +√

2,

√2 +

√2 +

√2, . . . exists and is

equal to 2.

HINT: The general term of this sequence satisfies the recursion relation an+1 =√

2 + an.

51. (i) Show in detail that {an} converges to the limit A precisely when every one of itssubsequences does.

(ii) Suppose that the sequences {an} and {bn} converge to the same limit, limn→∞

an = limn→∞

bn =

x. Show that any sequence {cn} whose terms are ak’s and bl’s also converges to x.

52. (i) Find lim sup and lim inf of the sequence

{1

n+ sin

nπ

2

}. Does this sequence have any

other cluster points?

8

(ii) Find lim sup and lim inf of the sequence {an} defined by

a1 =1

2, a2m =

a2m−1

2, a2m+1 =

1

2+ a2m.

53. (i) For any two real sequences {an} and {bn}, prove that

lim supn→∞

(an + bn) ≤ lim supn→∞

an + lim supn→∞

bn,

provided the sum on the right is not of the form ∞−∞.

(ii) Construct an example in which a strict inequality holds in (i).

54. Prove directly from the definition of the limit of a function that limx→a

[f(x) + g(x)] =

limx→a

f(x) + limx→a

g(x), provided the limits on the right-hand side exist.

55. Evaluate the limit limx→0

(1− cos x)2

sin2 x tan2 x.

56. For the following functions, do the left-hand and/or right-hand limits of f exist at x = 0?If either limit exists, what is its value?

(i) f(x) =

−1, x < 0,

0, x = 0,

1, x > 0.

(ii) f(x) = e1/x.

(iii) f(x) = sin1

x.

57. Show that the following functions are continuous:

(i) xn, for all x ∈ R.

(ii) 1/xn for x 6= 0.

(iii) z2 for all z ∈ C.

58. Suppose f is a real function defined on R which satisfies

limh→0

[f(x + h)− f(x− h)] = 0

9

for every x ∈ R. Does this imply that f is continuous.

59. Let f be a continuous real function defined on R, and let A, B, and C be the sets of allx ∈ R such that f(x) = 0, f(x) ≥ 0, and f(x) > 0, respectively. Show that A and B areclosed and C is open.

60. Give an example of a real continuous function defined on R such that

(i) the image of an open set is not open,

(ii) the image of a closed set is not closed.

61. Let f be a continuous real function on R.

(i) Is it necessarily true that f

(lim sup

n→∞xn

)= lim sup

n→∞f(xn)?

(ii) Is it true that x being a limit point of {xn} implies f(x) being a limit point of f(xn)?

(iii) Is it true that the inverse image under f of a convergent sequence is necessarily aconvergent sequence?

62. (i) Let f be continuous on an interval, I, and let f(x) = 0 when x is rational. Show thatf(x) = 0 for all x ∈ I.

(ii) Let f and g be continuous real functions defined on an interval, I. Let A ⊂ I be densein I. Show that f(A) is dense in f(I). If f(x) = g(x) for all x ∈ A, show that f(x) = g(x)for all x in I. (In other words, a continuous function is determined by its values on a densesubset of its domain.)

63. If f is a real continuous function defined on a closed set A ⊂ R, prove that there existsa continuous real function g on R such that g(x) = f(x) for all x ∈ A.

HINT: Let the graph of g be a straight line on each of the open intervals which constitutethe complement of A. This is a long and difficult problem, and to understand what the

difficulty is, imagine, say, A =∞∪

n=1[1/(2n + 1), 1/2n] ∪ [−1, 0] ∪ {1}.

64. (i) Show that if a continuous function on an interval takes on only a finite number ofvalues, it must be a constant.

10

(ii) Let f = p + g, where p is a polynomial of odd degree and g is a bounded continuousfunction on the line. Show that there is at least one solution of f(x) = 0.

HINT: p(x) attains arbitrarily large positive and negative values for large positive and neg-ative x, respectively.

65. Determine which of the following functions is uniformly continuous on the indicatedintervals.

(i) x3 on [−1, 1].

(ii) x3 on R.

(iii) 1/x2 on [1, 2].

(iv) 1/x2 on (0, 2).

66. (i) Let f be a real uniformly continuous function on a bounded set A ⊂ R. Prove thatf is bounded.

HINT: If it were not, you can find a sequence {xn} in A such that |f(xn+1)| > |f(xn)|+1. Finda convergent subsequence {xnk

} and derive a contradiction by showing that |xnk−xnl

| < 1/mfor large enough k and l, yet f (xnk

)− f (xnl) > 1.

(ii) Give an example to show that the conclusion of (i) is false if boundedness of A is omittedfrom the hypothesis.

67. (i) If f and g are uniformly continuous and bounded real functions on R, show that fgis uniformly continuous.

(ii) Give an example to show that the conclusion of (i) is false if boundedness of f and g isomitted from the hypothesis.

68. A uniformly continuous function of a uniformly continuous function is uniformly contin-uous. State this more precisely and prove it.

69. (i) Let K = {0} ∪ {1/n | n = 1, 2, 3, . . .}. Prove that K is compact directly from thedefinition, without using the Heine-Borel theorem.

(ii) Give an example of an open cover of the open interval (0, 1) which has no finite subcover.

70. Construct a compact set of real numbers whose limit points form a countable set.

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71. (i) Suppose f is a uniformly continuous real function defined on a set A ⊂ R. Provethat {f(xn)} is a Cauchy sequence in f(A) for every Cauchy sequence {xn} in A.

HINT: The proof is a bit similar to that of 66 (i).

(ii) Give an example to show that this is not the case if f is not uniformly continuous.

72. Let A be a dense subset of an interval, I, and let f be a uniformly continuous realfunction defined on A. Show that there exists a unique continuous function g, defined on I,such that g(x) = f(x) on A, by completing the following outline:

(i) Let x ∈ I−A. Use the results of problems 71 (i), 47, and 51 to prove that you can defineg(x) = lim

n→∞f(xn) for any sequence {xn} in A with xn → x, and that the value g(x) does

not depend on the choice of {xn}.

(ii) To show that g is continuous at any x ∈ I, let {xn} be an arbitrary sequence in I withxn → x. Use the result of (i) to show that, given ε > 0, for each xn, there is a yn ∈ A suchthat |xn− yn| < 1/n and |g(xn)− f(yn)| < ε/2. Show that also yn → x and use this to arguethat if n is large enough, |g(x)− g(xn)| < ε.

(iii) For uniqueness, use the result of problem 62 (ii).

73. Let a real function f on R be continuous, and let it satisfy the equation f(x + y) =f(x) + f(y) for all x, y ∈ R. Show that f(x) = cx for some constant x.

HINT: First, find the values of f(x) for rationals.

74. Provide an alternative proof of the Theorem: A continuous function on a compactset is uniformly continuous, by completing the details of the following argument. If K iscompact and f is not uniformly continuous on K, then for some ε > 0 there are sequences{xn} and {tn} in K such that |xn − tn| → 0 but |f(xn)− f(tn)| ≥ ε. Use the fact that anysequence in K must have a convergent subsequence to obtain a contradiction.

75. If E is a nonempty subset of C, define the distance from x ∈ C to E by

ρE(x) = infz∈E

|x− z|.

(i) Prove that ρE(x) = 0 if and only if x ∈ E (the closure of E).

(ii) Prove that ρE(x) is a uniformly continuous function on C by showing that

|ρE(x)− ρE(y)| ≤ |x− y|

12

for all x, y ∈ C.

HINT: ρE(x) ≤ |x− z| ≤ |x− y|+ |y − z|, so that

ρE(x) ≤ |x− y|+ ρE(y).

(iii) Suppose K and F are disjoint subsets of C, K is compact, F is closed. Prove that thereexists δ > 0 such that |z − w| > δ if z ∈ K, w ∈ F .

HINT: ρF is a continuous positive function on K.

(iv) Show that the conclusion of (iii) may fail for two disjoint closed sets if neither is compact.

76. Let I = [0, 1], the closed unit interval. Suppose f is a continuous function of I into I.Prove that f(x) = x for at least one x ∈ I.

77. Call a mapping from R to R open if f(A) is an open set whenever A is. Prove that everycontinuous open mapping is monotonic.

78. Let [x] denote the largest integer contained in x, that is, [x] is the integer such thatx−1 < [x] ≤ x. What kind of discontinuities do the functions [x] and x− [x] have? Describethem in as much detail as you can.

79. Find and classify the discontinuities of the following functions

(i) f(x) = e1/x + sin1

x,

(ii) f(x) =1

1− e−1/x.

80. Prove that

f(x) = limn→∞

[lim

m→∞(cos n!πx)2m

]=

{0, x irrational,

1, x rational.

What kind of discontinuities does the function f have?

81. Suppose a and c are real numbers, c > 0, and f is defined on [−1, 1] by

f(x) =

{|x|a sin(|x|−c), x 6= 0

0, x = 0.

13

Show the following statements:

(i) f is continuous if and only if a > 0.

(ii) f ′(0) exists if and only if a > 1.

(iii) f ′ is bounded if and only if a ≥ 1 + c.

(iv) f ′ is continuous if and only if a > 1 + c.

82. Let fn denote the n-th iterate of f , f1(x) = f(x), f2(x) = f(f1(x)), . . . , fn(x) =f(fn−1(x)). Express f ′n in terms of f ′. Show that if a ≤ |f ′(x)| ≤ b for all x, then an ≤|f ′n(x)| ≤ bn.

83. Let f be defined for all real x, and suppose that |f(x) − f(t)| ≤ (x − t)2 for all real xand t. Show that f is constant.

84. If

C0 +C1

2+

C2

3+ · · ·+ Cn−1

n+

Cn

n + 1= 0,

where C0, . . . Cn are real constants, prove that the equation

C0 + C1x + · · ·Cn−1xn−1 + Cnx

n = 0

has at least one real root between 0 and 1.

85. Suppose f is defined and differentiable for every x > 0 and f ′(x) → 0 as x →∞. Provethat f(x + 1)− f(x) → 0 as x →∞.

86. Suppose g is a real function on R with a bounded derivative (say |g′| ≤ M). Fix ε > 0and defined f(x) = x + εg(x). Prove that f is one-to-one if ε is small enough. Determinethe set of admissible values of ε as depending on M .

87. Suppose

(i) f is continuous for x ≥ 0,

(ii) f ′(x) exists for x > 0,

(iii) f(0) = 0,

14

(iv) f ′ is monotonically increasing.

Put

g(x) =f(x)

x, x > 0,

and prove that g is monotonically increasing.

HINT: Differentiate g and use the mean-value theorem.

88. Suppose f ′ is continuous on [a, b] and ε > 0. Prove that there exists a δ > 0 such that∣∣∣∣f(t)− f(x)

t− x− f ′(x)

∣∣∣∣ < ε

whenever 0 < |t− x| < δ, a ≤ x, t ≤ b.

HINT: Use the uniform continuity of f ′ and the mean-value theorem.

89. Let f be a continuous real function on R, of which it is known that f ′(x) exists for allx 6= 0 and that f ′(x) → 3 as x → 0. Does it follow that f ′(0) exists?

HINT: Use the mean-value theorem carefully to show that it does.

90. Suppose f is a real function on (−∞,∞). Call x a fixed point of f if f(x) = x.

(i) If f is differentiable and f ′(t) 6= 1 for all real t, show that f has at most one fixed point.

HINT: Mean-value theorem.

(ii) Show that the function f defined by

f(t) = t +1

1 + et

has no fixed point although 0 < f ′(t) < 1 for all real t.

(iii) However, if there is a constant A < 1 such that |f ′(t)| ≤ A for all real t, prove that afixed point x of f exists, and that x = limn→∞ xn, where x1 is an arbitrary number and

xn+1 = f(xn)

for n = 1, 2, 3, . . . .

(vi) Show that the process described in (iii) can be visualized by the zig-zag path

(x1, x2) → (x2, x2) → (x2, x3) → (x3, x3) → (x3, x4) → · · · .

15

91. Second Mean-Value Theorem: Let f and g be continuous on [a, b] and differentiableon (a, b), with g(a) 6= g(b). Prove that there exists a point c ∈ (a, b) such that

f(b)− f(a)

g(b)− g(a)=

f ′(c)

g′(c).

HINT: Apply the mean-value theorem to the function [f(b)− f(a)]g(x)− [g(b)− g(a)]f(x).

92. If f(x) = |x|3, compute f ′(x), f ′′(x) for all real x, and show that f ′′′(0) does not exist.

93. Suppose F ′′ is continuous in a neighborhood of x. Show that

limh→0

F (x + h) + F (x− h)− 2F (x)

h2= F ′′(x).

HINT: Replace x, f(x), and g(x) in problem 91 by t, F (x+t)+F (x−t), and t2, respectively,and let a = 0 and b = h. You can compute the remaining limit either directly or byL’Hospital’s rule.

94. Let a < b and

f(x) =

{(x− a)2(x− b)2, x ∈ [a, b]

0, otherwise.

Show that f is a continuously differentiable function that is non-zero exactly on the interval(a, b).

95. Let A be a closed subset of R. Construct a continuously differentiable real functiondefined on R that vanishes exactly on A.

96. Show that the function

f(x) =

{e−1/x2

, x 6= 0,

0, x = 0,

has all the derivatives at x = 0, and that they are all equal to 0 and continuous.

HINT: Use the fact that limt→∞

tαe−t = 0 for all α.

97. Suppose that f ≥ 0, f is continuous on [a, b], and

∫ b

a

f(x) dx = 0. Prove that f(x) = 0

for all x ∈ [a, b].

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98. Compute

limn→∞

1

n

(1 + sec2 π

4n+ sec2 2π

4n+ · · ·+ sec2 nπ

4n

).

Recall that sec x = 1/ cos x.

99. Let f and p be continuous and p(x) > 0 on [a, b]. Prove that there exists a ξ ∈ [a, b]such that ∫ b

a

f(x)p(x) dx = f(ξ)

∫ b

a

p(x) dx.

100. Prove that if the real-valued function f is integrable on the interval [a, b] then so isf 2. Using the identity (f + g)2 = f 2 + 2fg + g2, prove that the product of two integrablefunctions is integrable.

HINT: First show that |f 2(x) − f 2(t)| ≤ 2M |f(x) − f(t)|, where M = supx∈[a,b]

|f(x)|. If

P = {x0, . . . , xn} is a partition of [a, b], let

Mi(f) = supx∈[xi−1,xi]

f(x), mi(f) = infx∈[xi−1,xi]

f(x),

and deduce that if x, t ∈ [xi−1, xi], then

|f 2(x)− f 2(t)| ≤ 2M [Mi(f)−mi(f)].

101. Let y(x) be a continuously differentiable function on the interval [a, b]. Show that thelength of the curve (x, y(x)) for a < x < b is given by the expression∫ b

a

√1 + [y′(x)]2 dx

by completing the following outline:

Let a = x0 ≤ x1 ≤ . . . ≤ xn = b be any partition of the interval [a, b]. Let S be the polygonalcurve with corners at the points (xk, y(xk)) for k = 1, . . . , n.

(i) Let M = maxx∈[a,b]

|y′(x)|. Show that, given ε > 0, if ∆xi < ε/2M , then S approximates y(x)

to within ε.

17

(ii) Show that the length of the polygonal curve S equals

l(S) =n∑

k=1

√1 +

[y(xk)− y(xk−1)

xk − xk−1

]2

(xk − xk−1)

=n∑

k=1

√1 + [y′(ξk)]

2 (xk − xk−1) .

for some ξk ∈ [xk−1, xk], k = 1, . . . , n.

(iii) Let max |xk − xk−1| → 0 to complete the proof.

102. Let p and q be positive real numbers such that

1

p+

1

q= 1. (1)

Prove the following statements:

(i) If u ≥ 0 and v ≥ 0, then

uv ≤ up

p+

vq

q.

Equality holds if and only if up = vq.

HINT: Compute the areas between the curve y = xp−1 and the two coordinate axes, startingat the origin and ending at x = u and y = v, respectively. Use (1).

(ii) If f and g are integrable on [a, b], f ≥ 0, g ≥ 0, and∫ b

a

fp(x) dx =

∫ b

a

gq(x) dx = 1,

then ∫ b

a

f(x)g(x) dx ≤ 1.

(Problems 100 and 107 guarantee that all these functions are integrable.)

(iii) If f and g are integrable on [a, b], then∣∣∣∣∫ b

a

f(x)g(x) dx

∣∣∣∣ ≤ {∫ b

a

|f(x)|p dx

}1/p {∫ b

a

|g(x)|q dx

}1/q

. (2)

This is Holder’s inequality. When p = q = 2 it is called the Cauchy-Schwartz inequality.

18

103. For u integrable on [a, b] define

‖u‖2 =

{∫ b

a

|u(x)|2 dx

}1/2

.

Assume f , g, and h to be integrable on [a, b], and prove the triangle inequality

‖f − h‖2 ≤ ‖f − g‖2 + ‖g − h‖2

as a consequence of the Cauchy-Schwartz inequality from problem 102 (iii).

HINT: First square it.

104. With the notation as in problem 103, suppose f is integrable on [a, b] and ε > 0. Provethat there exists a continuous function g on [a, b] such that ‖f − g‖2 < ε.

HINT: For a suitable partition P = {x0, . . . , xn} of [a, b], define

g(t) =xi − t

∆xi

f(xi−1) +t− xi−1

∆xi

f(xi)

if xi−1 ≤ t ≤ xi. Argue that |f(t)− g(t)| < Mi −mi on [xi−1, xi], where

Mi = supx∈[xi−1,xi]

f(x), mi = infx∈[xi−1,xi]

f(x),

and that0 ≤ [f(t)− g(t)]2 ≤ 2M(Mi −mi),

where M = supx∈[a,b]

f(x).

105. Suppose f is continuously differentiable on [a, b], with f(a) = f(b) = 0, and∫ b

a

f 2(x) dx = 1.

Prove that ∫ b

a

xf(x)f ′(x) dx = −1

2,

and that ∫ b

a

[f ′(x)]2 dx ·∫ b

a

x2f 2(x) dx >1

4.

HINT: While it is easy to show ≥ in the last inequality, > requires you to solve a simpledifferential equation.

19

106. Show without an explicit variable change that∫ x/√

1+x2

0

dt√1− t2

=

∫ x

0

du

1 + u2.

What does this equation say about inverse trigonometric functions?

107. Prove the following that a continuous function of an integrable function is integrable.In particular, prove the Theorem: Suppose f is integrable on [a, b], m ≤ f ≤ M , φ iscontinuous on [m, M ] and h(x) = φ(f(x)) on [a, b]. Then h is integrable on [a, b].

HINT: Choose ε > 0 and 0 < δ < ε such that |φ(s)−φ(t)| < ε if |s− t| < δ and s, t ∈ [m, M ].Argue that for some partition P = {x0, . . . , xn} of [a, b],

S+(f, P )− S−(f, P ) < δ2. (3)

Let Mi and mi, and M∗i and m∗

i , are the suprema and infima of f and h on [xi−1, xi],respectively. Divide 1, . . . , n into two classes: i ∈ A if Mi −mi < δ, i ∈ B if Mi −mi ≥ δ.Show that, for i ∈ A, M∗

i −m∗i ≤ ε, and for i ∈ B, M∗

i −m∗i ≤ 2K, with K = max

x∈[m,M ]|φ(t)|.

Use (3) to show that

δ∑i∈B

∆xi < δ2,

and conclude thatS+(h, P )− S−(h, P ) ≤ ε(b− a) + 2Kδ.

108. Suppose f is bounded on [a, b], and f 2 is integrable. Does it follow that f is integrable?Does the answer change if we assume that f 3 is integrable?

HINT: The result of problem 107 should help.

109. If p ≥ q > 0, use the Cauchy-Schwartz inequality to prove that

logp

q≤ p− q√

pq.

110. Use the properties of the geometric progression to show that

x− x2

2+

x3

3− · · · − x2n

2n< log(1 + x) < x− x2

2+

x3

3− · · ·+ x2n+1

2n + 1

for 0 < x < 1 and any integer n ≥ 1.

20

HINT: First verify that log(1 + x) =

∫ x

0

du

1 + u.

111. Show that for n = 1, 2, 3, . . . , the number

Sn = 1 +1

2+

1

3+ · · ·+ 1

n− log n

is positive, that it decreases as n increases, and hence that the sequence of these numbersconverges to a limit, γ, between 0 and 1.

HINT: Verify that ∫ n+1

n

dx

x<

1

n<

∫ n

n−1

dx

x,

and use it to show that Sn is bounded between 0 and 1 and monotonically decreasing.

112. Evaluate the following limits

(i) limh→0

(1 + hx)1/h,

(ii) limn→∞

n(x1/n − 1) if x > 0,

(iii) limx→0

log(1 + x)

x.

(iv) limx→0

e− (1 + x)1/x

x.

(v) limn→∞

n

log n(n1/n − 1).

113. Suppose f(x)f(t) = f(x + t) for all real x and t.

(i) Assuming that f is differentiable and not zero, prove that

f(x) = ecx,

where c is a constant.

(ii) Prove the same thing assuming only that f is continuous.

HINT: In (ii), proceed as in problem 73.

114. Compute the arclength of the parabola y = x2/2 between the origin and the pointwhose abscissa is x = a > 0.

21

115. For positive integer n, prove that the integral∫

x2n+1ex2dx can be computed in terms

of elementary functions.

HINT: Find a suitable recursion formula. No need to compute the integral explicitly.

116. (i) Consider the indefinite integral∫dx√

ax2 + 2bx + c.

Depending on the sign of the coefficient a 6= 0 and the discriminant b2 − ac, show that thisintegral leads to log, arcsin, arccosh, or arcsinh. Describe the corresponding substitutionsand results in detail.

(ii) Show that the substitution t = 1/x transforms an integral of the type∫dx

x√

Ax2 + 2Bx + C

into an integral of the type discussed in part (i).

117. Derive all the solutions of the differential equation

y′ = 6yy′ + y′′′

that vanish at x → ±∞ together with all their derivatives.

HINT: At some convenient point, in order to integrate the equation once more, you shouldmultiply it by y′. You can also assume that y > 0. The solution is

y =1

2sech2 (x− x0)

2.

(You will get no points at all if you just plug this solution into the equation and check itworks!)

118. Compute the indefinite integrals

(i)

∫dx

(x2 − 1)2,

(ii)

∫x dx

(x2 + x + 1)2.

22

119. Let R(·, ·) denote an expression which is rational in both its arguments.

(i) Show that the integral

∫R(cos x, sin x) dx can be transformed into an integral of a

rational function of the variable t = tan x/2.

(ii) Show that the integral

∫R(cosh x, sinh x) dx can be transformed into an integral of a

rational function of the variable t = tanh x/2.

(iii) Show that the integral

∫R(x,

√1− x2) dx can be transformed into an integral of the

type discussed in part (i) by the substitution x = cos u.

(iv) Show that the integrals

∫R(x,

√x2 ± 1) dx can be transformed into integrals of the

type discussed in (ii) by the substitutions x = sinh u and x = cosh u, respectively.

120. Evaluate the integral ∫dx

a cos x + b sin x

as a function of cos x, sin x, a, and b.

121. (i) Show that the Fresnel integrals,

F1 =

∫ ∞

0

sin(x2

)dx, F2 =

∫ ∞

0

cos(x2

)dx,

converge.

(ii) Show that the integral ∫ ∞

0

2x sin(x4

)dx

converges even though its integrand becomes unbounded as x →∞.

HINT: Use appropriate substitutions.

122. Let f be a continuous real function on [0,∞), let

∫ ∞

a

f(x)

xdx converge for every a > 0,

and let limx→0

f(x) = L. Prove that ∫ ∞

0

f(αx)− f(βx)

xdx

23

converges for all positive α and β and has the value L logβ

α.

123. Show that

Γ(n) =

∫ 1

0

(log

1

x

)n−1

dx.

124. Let a < 0 and b > 0 and let F be bounded on [a, 0) and (0, b]. The Cauchy principalvalue integral of F on [a, b] is defined as

P∫ b

a

F (x) dx = limε→0+

[∫ −ε

a

F (x) dx +

∫ b

ε

F (x) dx

].

(i) Compute

P∫ 1

−1

dx

x.

(ii) If a < 0 < b and f is continuously differentiable on [a, b], show that

P∫ b

a

f(x)

xdx

exists.

HINT: Add and subtract f(0) in the numerator.

125. The integral

∫ ∞

a

f(x) dx is said to converge absolutely if

∫ ∞

a

|f(x)| dx converges. Prove

that if an integral converges absolutely, then it converges.

126. Let p(x) and q(x) be polynomials and let q(x) 6= 0 for x > a. Show that∫ ∞

a

p(x)

q(x)dx

converges absolutely if and only if their degrees satisfy the inequality deg(p) ≤ deg(q)− 2.

127. Show that ∫ ∞

0

cos x

1 + xdx =

∫ ∞

0

sin x

(1 + x)2dx,

and that one of these integrals converges absolutely, but the other does not.

24