Homework Spring 2025

1. Text section 1.1 number 4.

2. Text section 1.1 number 13.

3. What can be said about the truth value of Q when:

(a) P is false and P ⇒ Q is true?
(b) P is true and P ⇒ Q is true?
(c) P is true and P ⇒ Q is false?
(d) P is false and P ⇔ Q is true?
(e) P is true and P ⇔ Q is false?

4. Make a truth table for

(a) (P ∧ Q) ∨ (R ∧ ∼ S).

(b) [(P ∧ Q) ∨ (Q ∧ R)] ⇒ (P ∨ R).

5. Show that (P ⇒ Q) ⇒ R is
(a) equivalent to (P ∧ ∼ Q) ∨R
(b) not equivalent to P ⇒ (Q ⇒ R)

6. Consider the sentence: If x and y are even, then xy is even. (You may assume x, y ∈ Z.)
(a) Restate in words this sentence as a disjunction.
(b) State in words the negation of this sentence as a conjunction.

7. Text section 1.3 number 3 all 6 parts.

8. True or False. Explain each answer with a complete sentence. As appropriate include a counterexample.
(a) (∀x ∈ R)(∀y ∈ R)[(x < y ⇒ (∃w ∈ R)(x < w < y)].
(b) (∃x ∈ R)(∀y ∈ R)(x ≤ y)
(c) (∀y ∈ R)(∃x ∈ R)(x ≤ y)
(d) (∀n ∈ Z)(∃m ∈ Z)((n − 2)(m − 2) > 0)
(e) (∃n ∈ Z)(∀m ∈ Z)((n − 3)m < 1)

9. Find an useful denial for (∃!x)(A(x)).

10. Text section 1.3 number 12 all parts with explanation of why they are true or false. If parts are
equivalent to one another, include that as well.

11. Let P (x) and Q(x) be open sentences on a domain T . What do each of the following statements imply
about “∀x ∈ T, P (x) ⇒ Q(x)”? (Is it true or false or can’t tell?)
(a) “P (x) ∨ Q(x)” is false for all x ∈ T . (Note this is different than “∀x ∈ T, P (x) ∨ Q(x)” is false.)
(b) “Q(x)” is true for all x ∈ T .
(c) “P (x)” is false for all x ∈ T .
(d) “P (x) ∧ ( ∼ Q(x))” is true for some x ∈ T .
(e) “P (x)” is true for all x ∈ T .
(f) “( ∼ P (x)) ∧ ( ∼ Q(x))” is false for all x ∈ T .
12. Each part below is a definition of a mathematical term in bold. Each assigned part should be completed
for the statement in quotes.
(a) A sequence {an } converges to a limit L ∈ R (denoted {an } → L) if and only if “For all
positive real numbers ϵ, there exists a natural number N such that for all n ∈ N, if n > N , then
|an − L| < ϵ.”
(b) A sequence {an } diverges to ∞ (denoted {an } → ∞) if and only if “For all positive real numbers
M , there exists a natural number N such that for all n ∈ N, if n > N , then an > M .”
(c) A sequence {an } is Cauchy if and only if “For all positive real numbers ϵ, there exists a natural
number N such that for all n, m ∈ N, if n, m > N , then |an − am | < ϵ.”
(d) A real valued function f is continuous at a ∈ R if and only if “For all positive real numbers ϵ,
there exists a positive real number δ such that for all x ∈ R, if |x − a| < δ, then |f (x) − f (a)| < ϵ.”
(e) A set S ⊆ R is open if and only if “For all x ∈ S, there exists some positive real number ϵ such
that for every y ∈ R, we have |x − y| < ϵ implies y ∈ S.”
(f) A set S ⊆ R is closed if and only if “whenever a sequence {an } has every member in S and {an }
converges to a real number a, then a is also in S.”
i. State the given definition in symbols.
ii. State the negation of given definition in symbols.
iii. State the negation of given definition in words.
iv. Give a useful direct proof outline for the given definition.
v. Give a useful contradiction proof outline for the given definition.
vi. If possible, give a useful contrapositive proof outline for the given definition.

13. Text section 1.4 number 8.

(You do not need to prove the parts referenced in b.)

14. Let x, y, z ∈ Z. From definitions (not previous work) prove that:

(a) If x and y are odd, then xy is odd.
(b) If exactly one of x, y, z is odd, then xy + yz is even. (WLOG will be helpful here.)
(c) If xy is odd, then both x and y are odd.
(d) If x does not divide yz, then x does not divide z.
(e) If x ≥ 2 then x ∤ y or x ∤ (y + 1).
(f) Let x, y, z be positive, then xz divides yz if and only if x divides y.
(g) 2|x if and only if 2|x3 .

15. Using the technique of working backwards (see page 36) prove that:
(a) If a, b ∈ R+ , then ( ab + ab ) ≥ 2.
(b) If a, b ∈ R+ , then (a + b)( a1 + 1b ) ≥ 4.

16. Prove the following:

(a) The Triangle Inequality: For all x, y ∈ R, |x + y| ≤ |x| + |y|.
Hint: Uses cases depending on whether x and y share the same sign or not (but don’t forget 0).
(b) Disprove: For all x, y ∈ R, |x − y| ≤ |x| − |y|. (Explore how many different types of Counterex-
amples (CE) can you provide. (not part of the grade) )

Page 2
 (c) Using the Triangle Inequality prove that for all x, y ∈ R, |x| − |y| ≤ |x + y|.
Hint: “Add zero.” Also, we can extend this to get that for all x, y ∈ R, ||x| − |y|| ≤ |x + y|.
This extension is sometimes called the Reverse Triangle Inequality.
(d) Using the Triangle inequality that for all x ∈ R, if |x − 1| < 1, then |x + 1| < 3.
r
(e) Let a, b, x, y ∈ R and r ∈ R+ . If |x − a| < 2 and |y − b| < 2r , then |(x + y) − (a + b)| < r.
(f) Give an interpretation of the Triangle Inequality using geometry or distance.

17. Let x, y ∈ R. Prove that:

(a) If (x + 1)(x − 1) < 0, then x < 1.
(b) If x2 − 4x = y 2 − 4y and x ̸= y, then x + y = 4.
Note: Can you reword this where the conclusion is an “or”?
(c) If 3x4 + 1 ≤ x7 + x3 , then x > 0.
(d) If x < 0, then x3 − x2 y ≤ x2 y − xy 2 .
x+y x+y
(e) x < 2 if and only if 2 < y.
n N
(f) Prove that for natural numbers n, N that if n < N , then n+1 < N +1 .
√
18. Prove that 5 is irrational.

19. Text section 1.4 number 11bce.

20. Text section 1.5 number 12acd.

21. Disprove that:

(a) There is a real number x such that x6 + x4 − 2x2 + 1 = 0.
(b) For all positive integers n, n3 + n2 is odd.

22. Text section 1.6 number 8.

23. From definitions (not previous work) prove that:

(a) There exist integers m and n such that 15m + 12n = 3.
(b) There do not exist integers m and n such that 15m + 12n = 1.
(c) For every integer t, if there exist integer m and n such that 15m + 16n = t, then there exist
integers r and s such that 5r + 8s = t.
(d) There do not exist positive integers a and b such that a2 − b2 = 1.
(e) For all x ∈ R+ , there exists n ∈ N such that n > x if and only if for all ϵ ∈ R+ there exists n ∈ N
such that n1 < ϵ.

(f) Let b, L ∈ R. Prove if b ≥ L − ϵ for all positive ϵ, then b ≥ L.

24. Let a, b, c, d ∈ Z with a ̸= 0. Prove or Disprove:

(a) If a|bc, then a|b or a|c.
(b) If a|(b − c) and a|(c − d), then a|(b − d).
(c) If a ∤ bc, then a ∤ b and a ∤ c.

25. Let x, y ∈ R be given. Prove the following statements:

(a) If x + y is irrational, then either x or y is irrational.
(b) If x is rational and y is irrational, then x + y is irrational.

Page 3
 (c) There exist irrational numbers x and y such that x + y is rational.
(d) For every rational number z, there exist irrational numbers x and y such that x + y = z.

√ √ √
26. (a) Prove or disprove: For all positive real numbers a and b, a+b=
a + b.
√ √ √
(b) Prove or disprove: There exist positive real numbers a and b such that a + b = a + b.
√ √ √
(c) Fill in the blank: Let a, b ∈ R≥0 . Then a + b = a + b if and only if .
(d) Prove your answer to part c.

27. Let a, n ∈ Z.
(a) Prove that if a ≥ 2 and n ≥ 1 with a2 + 1 = 2n , then a is odd.
(b) Prove that there do not exist a ≥ 2 and n ≥ 1 such that a2 + 1 = 2n .

28. Text section 1.6 number 9cef ghj.

29. Text section 1.7 number 11acdef g.

30. For which integers n is 6 · 7n − 2 · 3n divisible by 4?

(a) Prove your claim by induction.
(b) Can you give a direct proof?
(c) Prove by induction that 7|(32n+1 + 2n+2 ) for all integers n ≥ 0.

31. Prove the following by induction: (See page 129 − 130 for the definition of fn .)
n
2i = 2n+1 − 2.
P
(a) For all n ∈ N,
i=1
1 2 n 1
(b) For all n ∈ N, 2! + 3! + ··· + (n+1)! =1− (n+1)! .
(c) Let s ∈ R, t ∈ R − {0, 1}. Prove for all n ∈ N that
n−1
X s(tn − 1)
sti = (Finite geometric sum)
t−1
i=0

(d) For all n ∈ N, 8|(52n − 1).

(e) Let a ∈ Z. For all n ∈ N, if 2|3n a, then 2|a. (You may not use Euclid’s lemma here.)
(f) Let p be a prime number and a be a natural number. If p|an , then p|a for all natural numbers
n ≥ 2. (You may use Euclid’s lemma (1.8.3) here as needed.)
(g) For all natural number n ≥ 5, (n + 1)! > 2n+3 .
n 2
i −1
= n+1
Q
(h) For all natural numbers n ≥ 2 that i2 2n .
i=2
(i) For all n ∈ N, f4n is a multiple of 3.

32. Prove by induction that for every integer n ≥ 2: If x1 , x2 , . . . , xn are n real numbers with x1 ·x2 ·· · · xn =
0, then at least one of x1 , x2 , . . . , xn is 0.

33. Text section 2.4 number 12cdeg.

34. Complete the following by strong induction. (See page 129 − 130 for the definition of fn .)

Page 4
 (a) Let t1 = 1, t2 = 2, t3 = 3 and for all n ≥ 4 let tn = tn−1 + tn−2 + tn−3 . Prove that tn < 2n for all
n ∈ N.
(b) Every natural number n ≥ 23 can be written as 3s + 4t where s, t ∈ Z with s ≥ 3 and t ≥ 2.
(c) For all n ∈ N, f1 + f2 + f3 + · · · + fn = fn+2 − 1.
(d) For all n ∈ N, fn+6 = 4fn+3 + fn .

35. Text section 2.5 number 14df .

36. Let U be a set and let P be an open statement with domain U . Let A = {x ∈ U : P (x) is true}, i.e.,
the truth set of P . Let a ∈ U be fixed. Are the following true or false? Explain your answers.
(a) If a ∈ A, then we may deduce that P (a) is true.
(b) If P (a) is true, then we may deduce that a ∈ A.
(c) If ∼ P (a) is true, then we may deduce that a ̸∈ A.
(d) If a ̸∈ A, then we may deduce that ∼ P (a) is true.

37. Are the following true or false? Explain your answers. (A, B, C are arbitrary sets where needed.)

(a) 4 ⊆ {4}. (g) For every set A, ∅ ∈ A. (m) If A ⊆ B and B ∈ ℘(C),

(b) {3} ∈ {3, {3, 7}, {7}}. (h) For every set A, {∅} ⊆ A. then A ∈ ℘(C).
(c) {3} ⊆ {3, {3, 7}, {7}}. (i) If x ∈ A, then x ∈ ℘(A). (n) {∅} ∈ ℘({∅, {∅}}).
(d) ∅ ∈ ∅. (j) If x ∈ A, then {x} ∈ ℘(A).
(o) {∅} ⊆ ℘({∅, {∅}}).
(e) {∅} ⊆ {∅, {∅}}. (k) If x ∈ A, then {x} ⊆ ℘(A).
(f) {∅} ∈ {∅, {∅}}. (l) For every set A, ∅ ∈ ℘(A). (p) {{∅}} ⊆ ℘({∅, {∅}}).

38. Let A and B be sets. Prove that

(a) If A ⊆ B, then ℘(A) ⊆ ℘(B).
(b) If ℘(A) ⊆ ℘(B), then A ⊆ B.
I know this is a Theorem in the text, please do your best to attempt to prove this without using
the Theorem.
(c) Can you extend your proof of the above statements when subset is changed to equality? Explain.

39. Text section 2.1 number 19abd.

40. Let A and B be two nonempty disjoint subsets of a universe S. If x ∈ S, are the following true or
false? Explain.
(a) It’s possible that x ∈ A ∩ B.
(b) If x ∈ A, then x ̸∈ B.
(c) If x ̸∈ A, then x ∈ B.
(d) It’s possible that x ̸∈ A and x ̸∈ B.
(e) For every nonempty set C ⊆ S, either x ∈ A ∩ C or x ∈ B ∩ C.
(f) There exists a nonempty set C ⊆ S, such that x ∈ A ∪ C and x ∈ B ∪ C.

41. Let A, B, C, D be sets. Prove that

Page 5
 (a) A ∩ C ⊆ B ∩ C if and only if C − B ⊆ C − A.
(b) If A ∩ B = ∅, then A = (A ∪ B) − B.
(c) A ̸= B if and only if (A − B) ∪ (B − A) ̸= ∅.
(d) If A ⊆ B, then A − C ⊆ B − C.
(e) If C ⊆ A, D ⊆ B, then D − A ⊆ B − C.
(f) If A ∪ B ⊆ C ∪ D, A ∩ B = ∅, and C ⊆ A, then B ⊆ D.
(g) (A × B) ∩ (C × D) = (A ∩ C) × (B ∩ D).
(h) A ∪ (B − C) = (A ∪ B) − (C − A).
(i) (A − B) ∪ (A − C) = A − (B ∩ C)

42. Demonstrate the following are false (provide a counterexample!): (A, B, C, D represent sets.)
(a) A − (B − C) = (A − B) − C.
(b) ℘(A) − ℘(B) ⊆ ℘(A − B).
(c) (A × B) ∪ (C × D) = (A ∪ C) × (B ∪ D).
(d) If A ∩ B = A ∩ C then B = C.
(e) If A ∪ B = A ∪ C then B = C

43. Let A, B, C be sets. Prove that “if A ∩ B = A ∩ C and A ∪ B = A ∪ C, then B = C” is true:

(a) Directly.
(b) By contrapositive.

44. Prove or disprove each of the following:

(a) Let A, B, C be sets. Let C be non-empty. For all proper subsets A of C, there exists a non-empty
subset B of C such that A ∪ B = C and A ∩ B = ∅.
(b) Let A, B, C, D be sets. If A ⊆ C, B ⊆ D, and A ∩ B = ∅, then C ∩ D = ∅.
(c) Let A, B, C, D be sets. If A ⊆ C, B ⊆ D, and C ∩ D = ∅, then A ∩ B = ∅.
(d) Let A, B be sets. If A − B = B − A, then A − B = ∅.
(e) Let A be a set. If A − B = ∅ for all sets B, then A = ∅.
(f) Let A be a set. If A ∩ B = ∅ for all sets B, then A = ∅.
(g) Let A be a set. If A ∪ B ̸= ∅ for all sets B, then A ̸= ∅.

45. Text section 2.2 number 12.

46. Text section 2.2 number 20bcef .

47. Let R be a relation from A to B and S be a relation from B to C.

(a) Prove that Dom(R−1 ) = Rng(R).
(b) Prove that Dom(S ◦ R) ⊆ Dom(R).
(c) Demonstrate that Dom(S ◦ R) = Dom(R) may be false.

48. Text section 3.1 number 17aef g.

49. Prove the given relations are equivalence relations and find the requested classes.

Page 6
 −3
(a) Define the relation V on R by xV y if and only if x = y or xy = 1. Find 4 and 0.
(b) Define the relation Q on the set Z × (Z − {0}) by (a, b)Q(c, d) if and only if ad = bc. Find (1, 3)
and (2, 1).

50. Let A = {a, b, 5, 11, −2, M } Define the relation R on ℘(A) by XRY if and only if X and Y have the
same number of elements.
(a) List {M } and {a, b, 11, −2, M }.
(b) How many elements are in A?

51. Prove the following:

(a) R is reflexive on A if and only if IA ⊆ R.
(b) R is symmetric on A if and only if R−1 = R.
(c) R is transitive on A if and only if R ◦ R ⊆ R.
(d) If R is symmetric and transitive on A and A ⊆ Dom(R), then R is reflexive on A.

52. Text section 3.2 number 19acd.

53. Let R be a relation on A.

(a) Prove that R ∪ R−1 is symmetric.
(b) Prove that R ∩ R−1 is symmetric.
(c) Repeat the above questions for the other definitions: reflexive, antisymmetric, transitive. Are
these new statements True or False?
(d) Prove that if R is a partial order on A, then R−1 is also a partial order on A.

54. Let R be the relation on N defined by (a, b) ∈ R if and only if there exists a k ∈ Z with k ≥ 0 such
that b = 2k a. Prove that R is a partial order. Does R have the comparability property?

55. Text section 3.5 number 9a.

56. Text section 3.5 number 22a.

57. Text section 4.1 number 10.

58. Let f : A → B be a function. Define the relation T on A by xT y if and only if f (x) = f (y). Prove
that T is an equivalence relation on A.

59. Text section 4.1 number 19abc.

60. Text section 4.2 number 6.

61. Text section 4.2 number 19abcdef .

62. Let f (x) = x2 + 3x − 5 be a real valued function.

(a) Show that f is not injective.
(b) Find all pairs r1 , r2 of real numbers such that f (r1 ) = f (r2 ).
(c) Show that f is not surjective.
(d) Find the interval S such that S = {s ∈ R | s ̸= f (x) ∀x ∈ R}.

Page 7
 3
(e) Give the largest restrictions of the domain that is an interval that contains 2 and the largest
codomain that make f bijective.

63. Prove that all degree 1 polynomials on R are bijective.

64. Let f : A → B and g : B → C be functions. Prove the following:

(a) If f and g are injective, then g ◦ f is injective.
(b) If f and g are surjective, then g ◦ f is surjective.
(c) If g ◦ f is injective, then f is injective.
(d) If g ◦ f is surjective, then g is surjective.

65. Let f : A → B and g : B → C be functions. Prove or disprove the following:

(a) If f and g are bijective, then g ◦ f is bijective.
(b) If g is surjective, then g ◦ f is surjective.
(c) If g is injective, then g ◦ f is injective.
(d) There exists an f that is not surjective with g ◦ f is surjective.
(e) There exists an f that is not injective with g ◦ f is injective.
(f) If g ◦ f is injective, then g is injective.

66. Let f : A → B, g : B → C, and h : B → C be functions. Prove or disprove the following:

(a) If g ◦ f = h ◦ f then g = h.
(b) If f is injective, then g ◦ f = h ◦ f ⇒ g = h.
(c) If f is surjective, then g ◦ f = h ◦ f ⇒ g = h.

67. Let f : A → B and g : B → C be functions and let g ◦ f be bijective. Prove f is surjective if and only
if g is injective.

68. Text section 4.3 number 13.

69. Text section 4.3 number 15bcef hi.

70. Text section 4.5 number 7.

71. Text section 4.5 number 8.

72. Let f : A → B, C ⊆ A and E ⊆ B.

(a) Prove that f (f −1 (E)) ⊆ E.
(b) Demonstrate that E ⊆ f (f −1 (E)) need not be true.

Page 8
 (c) Prove that E ∩ Rng(f ) ⊆ f (f −1 (E)).
(d) Prove that C ⊆ f −1 (f (C)).
(e) Demonstrate that f −1 (f (C)) ⊆ C need not be true.
(f) Prove that f −1 (f (C)) ⊆ C if and only if f (A − C) ⊆ B − f (C).

73. Let f : A → B, C, D ⊆ A and E, F ⊆ B. Prove that

(a) f (C) ⊆ E if and only if C ⊆ f −1 (E).
(b) f (C) − f (D) ⊆ f (C − D).
(c) if f is injective, then f (C − D) ⊆ f (C) − f (D).
(d) f −1 (E) − f −1 (F ) = f −1 (E − F ).

74. Let f : A → B.
(a) Prove that f is injective if and only if f (X) ∩ f (Y ) = f (X ∩ Y ) for all X, Y ⊆ A.
(b) Let X ⊆ A and Y ⊆ B. Prove that if f is a bijection, then f (X) = Y if and only if X = f −1 (Y ).
(c) Prove that f is injective if and only f −1 ({y}) has no more than 1 element for all y ∈ B.
(d) Prove that f is surjective if and only f −1 ({y}) has at least 1 element for all y ∈ B.

75. Text section 4.5 number 17abd.

76. Use the definition of convergence to complete the following:

4n
(a) Prove that the sequence { 3n−2 } converges to 34 .
1
(b) Prove that the sequence {−6 + n2
} converges to −6.
n
(c) Prove that the sequence { n+4 } converges to 1.

77. Use the definition of diverges to infinity to complete the following:

(a) Prove that the sequence {n7 } diverges to infinity.
√
(b) Prove that the sequence { 3 4n} diverges to infinity.

78. Use the definition of diverges (limit does not exist) to complete the following:
(a) Prove that the sequence {(−1)n } does not have a limit both directly from definition and by
contradiction.
(b) Prove that the sequence {(−3)n } does not have a limit.
√
(c) Prove that the sequence {(−1)n+1 3 n} does not have a limit.

79. Using contradiction or the negation the definition of convergence (to L), complete the following:
4n
(a) Prove that { 3n−2 } does not converge to 2.
5n
(b) Prove that { n+1 } does not converge to 2.

Page 9
80. Let {cn } be a sequence of real numbers.
(a) Let M ∈ R be fixed. Prove that if cn = M for all n ∈ N , then {cn } → M .
(b) Let {cn } ⊆ N and K ∈ N. Describe all possible sequences {cn } that converge to K.
Reminder {cn } ⊆ N means that cn ∈ N for all n ∈ N.

81. Let a, b, r, s ∈ R Prove from definition that if {an } → a and {bn } → b, then
(a) {an + bn } → a + b
(b) {an − bn } → a − b
(c) {−an } → −a
(d) {ran } → ra
(e) {ran + sbn } → ra + sb
(f) {|an |} → |a|
(g) {an bn } → ab (Hint: Remember to consider limiting to 0 separately.)

82. Let a, b ∈ R − {0} Prove that if {an } → a and {bn } → b and an ̸= 0 ∀n ∈ N, then
|a|
(a) there exists an N ∈ N such that if n ≥ N , then |an | > 2 .
(b) { abnn } → ab .

83. Suppose the sequence {an } converges to a > 1. Prove there exists an index N such that for all n > N ,
an > 1. (Note: 1 isn’t special in this question. We can change 1 to an arbitrary fixed real number
and this still works. We can also change the inequality to <, ≤, or ≥ in both places for a similar
question.)

84. Let {xn } → L and {yn } → M .

(a) Prove that if xn ≤ yn for all n ∈ N, then L ≤ M .
(b) Use (a) or prove directly that if xn ≥ 0 for all n ∈ N, then L ≥ 0.
(c) Prove or disprove that if xn < yn for all n ∈ N, then L < M .

85. Text section 4.6 number 11.

86. Text section 4.6 number 13.

87. Let {xn } be a convergent sequence and B ∈ R. Prove that if there exists N ∈ N with xn ≤ B for
all n > N , then lim xn ≤ B. (Could we change both ≤ signs to < signs and still have a true statement?)

88. Let {xn } → 0.

(a) Prove that if {yn } is bounded, then {xn yn } → 0.
(b) Provide a counterexample when {yn } is unbounded.

89. True or False. Justify your answer with at least one sentence or a well-explained counterexample if
appropriate.

Page 10
 (a) If {an + bn } converges , then {an } and {bn } both converge.
(b) If a sequences is bounded, then it must be convergent.
(c) Assume lim an exists. If lim an < 0, then an < 0 for all n ∈ N.
(d) Assume lim an exists. If an < 0 for all n ∈ N, then lim an < 0.
(e) If sequences {an }, {bn }, {cn }, . . . , {zn } converge to a, b, c, . . . z ∈ R respectively, then {an bn cn · · · zn }
converges to abc · · · z.

90. Prove all degree 1 polynomials are continuous on R:

(a) using the sequence definition.
(b) using the ϵ-δ definition.

91. Let f (x) = 3x2 − 20, a function on R.

(a) Prove that f is continuous at −3 using sequences.
(b) Prove that f is continuous at 2 using sequences.

92. Let g(x) = −2x2 + 7, a function on R.

(a) Prove that g is continuous at −1 using the ϵ-δ definition.
(b) Prove that g is continuous at 3 using the ϵ-δ definition.

93. Prove each of the following using the ϵ-δ definition of continuity.
√
(a) That f (x) = x is continuous at a = 9.
1
(b) That f (x) = x is continuous at a = 4.
(c) That f (x) = x3
is continuous at a = −2.

 −4 : x < −2
94. Define g a function on R by g(x) = x : −2 ≤ x ≤ 2
3 :x>2


(a) Prove that g is discontinuous at 2 using sequences.

(b) Prove that g is discontinuous at −2 using sequences.
(c) Prove that the lim g(x) does not exist with the sequences definition.
x→2


 x − 1 : x < −1
95. Define f a function on R by f (x) = x : −1 ≤ x ≤ 1
x+2 :x>1


(a) Prove that f is discontinuous at 1 using the ϵ-δ definition.

(b) Prove that f is discontinuous at −1 using the ϵ-δ definition.
(c) Prove that the lim f (x) does not exist with the ϵ-δ definition.
x→1

96. Let a ∈ I where I is a non-empty open interval of real numbers. Let f and g be functions defined on
I. Prove each of the following using the sequential definition of continuity:

Page 11
 (a) If f and g are continuous at a, then f + g is continuous at a.
(b) If f and g are continuous at a, then f − g is continuous at a.
(c) If f and g are continuous at a, then f g is continuous at a.
(d) Let r ∈ R. If f continuous at a, then rf is continuous at a.
(e) Let r, s ∈ R. If f and g are continuous at a, then rf + sg is continuous at a.

97. Let a ∈ I where I is a non-empty open interval of real numbers. Let f and g be functions defined on
I. Prove each of the following using the ϵ-δ definition of continuity:
(a) If f and g are continuous at a, then f + g is continuous at a.
(b) If f and g are continuous at a, then f − g is continuous at a.
(c) If f and g are continuous at a, then f g is continuous at a.
(d) Let r ∈ R. If f continuous at a, then rf is continuous at a.
(e) Let r, s ∈ R. If f and g are continuous at a, then rf + sg is continuous at a.

98. Let b, L ∈ R. Prove if b ≥ L − ϵ for all positive ϵ, then b ≥ L. (Note that this isn’t necessarily directly
related to the definitions in this section, but is a similar concept.)

99. Text section 4.7 number 9abc.

100. True or False. Justify your answer with at least one sentence or a well-explained counterexample if
appropriate.
(a) If f + g is continuous at a ∈ R, then f and g are both continuous at a also.
(b) If f g has a limit as x approaches a, then f and g both have limits as x approaches a.
(c) If f is continuous at a ∈ R, then f must be continuous at all points in an interval around a.

101. Let A, B ⊆ R. Prove the following:

(a) If A and B is bounded above, then A ∪ B is bounded above.
(b) If A and B is bounded below, then A ∪ B is bounded below.
(c) Why would we have to be more careful if we replaced ∪ with ∩ in the two previous parts?
(d) If A is bounded below, then AC is not bounded below. (Hint: What is A ∪ AC ?)
(e) If A is bounded above, then AC is not bounded above. (Hint: What is A ∪ AC ?)
(f) If sup(A) exists, then it is unique.
(g) If inf(A) exists, then it is unique.

102. Let A, B ⊆ R with A ⊆ B. Prove the following:

(a) If B is bounded above, then A is bounded above.
(b) If B is bounded below, then A is bounded below.
(c) If sup(A) and sup(B) both exist, then sup(A) ≤ sup(B).
(d) If inf(A) and inf(B) both exist, then inf(A) ≥ inf(B).

103. Let A ⊆ R and M be an upper bound for A, where M ∈ R. Prove the following:

Page 12
 (a) For all y ∈ R, if M < y, then y is an upper bound for A.
(b) If M ∈ A, then M = sup(A).
(c) Redo this question for lower bound and inf(A).

104. For each for the following sets find inf(S) and sup(S) and prove your answers are correct.
(a) Let S = {x ∈ R | x2 − x < 0}.
(b) Let S = {x ∈ Q | x2 < 5}.
(c) Let S = {x ∈ R | x > −7}.
(d) Let S = {x ∈ R | x < π}.
(e) Repeat the previous two parts with the domain as Q rather than R.

105. True or False. Justify your answer with at least one sentence or a well-explained counterexample if
appropriate.
(a) For a bounded nonempty set of real numbers, S, inf(S) ≤ sup(S).
(b) Every nonempty set of real numbers that is bounded below has a smallest member.
(c) If S is a set of negative real numbers, then sup(S) ≤ 0.

106. Text section 7.1 number 16.

107. Prove that S ⊆ R has one element if and only if inf(S) = sup(S).

108. Let S be a (non-empty) bounded set of real numbers. Prove there are sequences (most likely distinct!)
in S that converge to sup S and inf S.

109. Let S be a bounded set of real numbers. Let α = inf(S).

(a) Define A = {x ∈ R | x = 2y ∃y ∈ S}. Prove that inf(A) = 2α.
(b) Define A = {x ∈ R | x = 2y ∃y ∈ S}. Prove that sup(A) = 2 sup(S).
(c) Repeat the previous parts with a different constant −1, 21 , 3, or similar.
n−1
(d) Let S = {x ∈ R | x = n+1 ∃n ∈ N}. Find inf(S) and sup(S) and prove your answers are correct.

110. Text section 7.1 number 20bcd.

Page 13