Properties of real numbers - order, completeness, least upper bound property, Cauchy sequences - Question Bank

1. If a sequence (x_n) is a Cauchy sequence, then for any ε > 0, there exists an integer N such that for all n, m > N, the distance |x_n - x_m| is less than:
A) 1/N
B) ε
C) ε/2
D) Nε
2. Which property guarantees that the set of integers Z is not complete?
A) Z is not bounded above.
B) Z is not bounded below.
C) Z does not satisfy the least upper bound property for all its subsets.
D) Z does not satisfy the Archimedean Property.
3. The statement 'For any a, b ∈ R, either a < b, a = b, or a > b' is the definition of the:
A) Transitive Property
B) Addition Property
C) Trichotomy Property
D) Multiplication Property
4. Consider the set G = {n / (n+1) | n ∈ N}. What is the greatest lower bound (infimum) of G?
A) 0
B) 1/2
C) 1
D) Does not exist
5. If a sequence (x_n) is a Cauchy sequence, then for any M > 0, there exists an integer N such that for all n > N, |x_n| < M. This means the sequence is:
A) Convergent
B) Divergent
C) Bounded
D) Monotonic
6. The least upper bound property of real numbers is also known as the:
A) Monotone Convergence Theorem
B) Bolzano-Weierstrass Theorem
C) Completeness Axiom
D) Intermediate Value Theorem
7. Which property is essential for proving that every convergent sequence of real numbers is bounded?
A) Archimedean Property
B) Completeness Property
C) Order Property
D) Cauchy Sequence Criterion
8. If a sequence (x_n) converges, then the sequence (x_n / n) must:
A) Converge to 0
B) Diverge
C) Converge to 1
D) Be monotonic
9. Consider the set F = {x ∈ R | x² ≥ 4}. This set is:
A) Bounded above and below
B) Bounded above but not below
C) Bounded below but not above
D) Neither bounded above nor below
10. The property that for any two distinct rational numbers r1 and r2, there exists an irrational number 'i' such that r1 < i < r2, is known as:
A) Density of Rational Numbers
B) Density of Irrational Numbers
C) Completeness Property
D) Archimedean Property
11. A sequence (x_n) is Cauchy if and only if:
A) It is monotonic and bounded.
B) It converges to some real number.
C) It is bounded.
D) It is eventually constant.
12. If sup(S) = s, then for any ε > 0, the interval (s - ε, s + ε) must contain:
A) At least one element of S
B) All elements of S
C) No elements of S
D) Only elements of S greater than s
13. Which statement about the set of natural numbers N is FALSE?
A) N is bounded above.
B) N is bounded below.
C) N has a least element.
D) N has a greatest element.
14. The completeness property of the real numbers is what distinguishes them from:
A) Complex numbers
B) Rational numbers
C) Integers
D) Natural numbers
15. If a sequence (x_n) is monotonic and bounded, then it:
A) Must be a Cauchy sequence.
B) May or may not be a Cauchy sequence.
C) Must diverge.
D) Must be unbounded.
16. What is the infimum of the set E = {x ∈ R | x > 5}?
A) 4
B) 5
C) 6
D) Does not exist
17. The property that for any a, b ∈ R, if a < b, then a + c < b + c for any c ∈ R is called the:
A) Multiplication Property of Order
B) Addition Property of Order
C) Trichotomy Property
D) Transitive Property
18. If (x_n) and (y_n) are both Cauchy sequences, then the sequence (x_n + y_n) is:
A) Always convergent
B) Always bounded
C) Always a Cauchy sequence
D) Not necessarily Cauchy
19. Which of the following is a direct consequence of the Archimedean Property?
A) The existence of transcendental numbers.
B) The fact that the interval (0, 1) contains infinitely many rational numbers.
C) For any positive real number ε, there exists a natural number n such that 1/n < ε.
D) Every real number can be expressed as a limit of a Cauchy sequence.
20. The set of all upper bounds of a non-empty set S, which is bounded above, has:
A) No minimum element
B) A minimum element which is the supremum of S
C) A maximum element which is the supremum of S
D) No specific relation to the supremum of S
21. Consider the sequence x_n = (-1)^n / n. Is this sequence a Cauchy sequence?
A) Yes, because it converges to 0.
B) No, because it alternates signs.
C) Yes, because it is bounded and oscillates around 0.
D) No, because it is not monotonic.
22. If a set S is bounded above by 'u' and bounded below by 'l', then:
A) sup(S) ≥ u and inf(S) ≤ l
B) sup(S) ≤ u and inf(S) ≥ l
C) sup(S) > u and inf(S) < l
D) sup(S) < u and inf(S) > l
23. Which property ensures that for any real number x, there exists a natural number n such that n > x?
A) Completeness Property
B) Order Property
C) Density Property
D) Archimedean Property
24. The set of real numbers R is complete. This means:
A) Every subset of R is bounded.
B) Every non-empty subset of R that is bounded above has a least upper bound in R.
C) Every sequence in R converges.
D) There are no gaps in the real number line.
25. If a sequence (x_n) is a Cauchy sequence, which of the following must be true?
A) x_n = 0 for all n > N
B) lim (x_{n+1} - x_n) = 0
C) lim x_n = 0
D) x_n approaches infinity
26. What is the least upper bound of the set D = {x ∈ R | 0 < x < 1}?
A) 0
B) 0.5
C) 1
D) Does not exist
27. Which property states that if a < b and b < c, then a < c for real numbers a, b, and c?
A) Trichotomy Property
B) Addition Property
C) Transitive Property
D) Multiplication Property
28. Consider the set C = {1/n | n ∈ N}. What is the greatest lower bound (infimum) of C?
A) 1/2
B) 1
C) 0
D) Does not exist
29. If a sequence (x_n) converges to L, then it implies that (x_n) is:
A) Always monotonic.
B) Always a Cauchy sequence.
C) Always bounded, but not necessarily Cauchy.
D) Always unbounded.
30. The completeness property of real numbers is crucial for establishing the existence of:
A) The additive inverse of every real number.
B) The multiplicative inverse of every non-zero real number.
C) The limit of every convergent sequence.
D) The square root of every positive real number.
31. Let S be a non-empty subset of R. If 's' is the least upper bound of S, then which statement is always true?
A) For every ε > 0, there exists x ∈ S such that s - ε < x ≤ s.
B) For every ε > 0, there exists x ∈ S such that s + ε < x ≤ s.
C) s is an element of S.
D) s is greater than all elements of S.
32. If a sequence (x_n) is NOT a Cauchy sequence, then:
A) It must be unbounded.
B) It must diverge.
C) There exists an ε > 0 such that for any N, there are n, m > N with |x_n - x_m| ≥ ε.
D) It must be monotonic.
33. The property that for any two distinct real numbers a and b, there exists a rational number r such that a < r < b, is known as:
A) Completeness Property
B) Archimedean Property
C) Density of Rational Numbers
D) Order Property
34. Consider the set B = {x ∈ R | x² < 2}. What is the least upper bound of B?
A) 1
B) √2
C) 2
D) Does not exist
35. Which of the following statements about a sequence (x_n) is equivalent to (x_n) being a Cauchy sequence?
A) (x_n) is bounded and monotonic.
B) (x_n) converges.
C) (x_n) has a convergent subsequence.
D) (x_n) eventually becomes constant.
36. If (x_n) is a Cauchy sequence, then for any ε > 0, there exists N such that for all n ≥ N, the term x_n is within what range?
A) (L - ε, L + ε) for some L
B) (x_N - ε, x_N + ε)
C) (x_m - ε, x_m + ε) for some m
D) (-ε, ε)
37. The property that every non-empty set of real numbers bounded below has a greatest lower bound (infimum) is a consequence of:
A) The Archimedean Property
B) The Completeness Property applied to the set of additive inverses
C) The Order Axioms
D) The Density Property of Rational Numbers
38. Let S be a non-empty set of real numbers. If S has an upper bound 'u', then the least upper bound (supremum) of S, denoted by sup(S), satisfies:
A) sup(S) > u
B) sup(S) < u
C) sup(S) ≤ u
D) sup(S) = u + 1
39. Which property ensures that the union of two sets, each bounded above, is also bounded above?
A) Archimedean Property
B) Trichotomy Property
C) Completeness Property
D) Transitive Property of Inequalities
40. What is the supremum of the set S = {1 - 1/n | n ∈ N}?
A) 0
B) 1/2
C) 1
D) 2
41. Consider the sequence x_n = 1/n for n ∈ N. Is this sequence a Cauchy sequence?
A) Yes, because it converges to 0.
B) No, because the terms are always positive.
C) Yes, because |1/n - 1/m| can be made arbitrarily small.
D) No, because it is decreasing.
42. If a sequence (x_n) is a Cauchy sequence, then it must be:
A) Monotonically increasing
B) Monotonically decreasing
C) Bounded
D) Divergent
43. What is a key characteristic of Cauchy sequences in the context of real numbers?
A) Every Cauchy sequence converges.
B) Every convergent sequence is a Cauchy sequence.
C) A sequence is Cauchy if and only if it is convergent.
D) Cauchy sequences are always monotonic.
44. A sequence (x_n) of real numbers is called a Cauchy sequence if:
A) It converges to a limit L.
B) For every ε > 0, there exists an integer N such that |x_n - x_m| < ε for all n, m > N.
C) It is bounded.
D) It is monotonic.
45. Which of the following is NOT a property of the order relation '<' on the set of real numbers R?
A) Trichotomy: For any a, b ∈ R, exactly one of a < b, a = b, or b < a is true.
B) Transitivity: For any a, b, c ∈ R, if a < b and b < c, then a < c.
C) Addition: For any a, b, c ∈ R, if a < b, then a + c < b + c.
D) Multiplication: For any a, b, c ∈ R, if a < b and c > 0, then ac < bc, and if c < 0, then ac > bc.
46. The Archimedean Property is fundamental for proving which concept related to real numbers?
A) The existence of irrational numbers
B) The convergence of sequences
C) The density of rational numbers in real numbers
D) The completeness of the real number line
47. Which property of real numbers states that for any positive real number ε, there exists a natural number n such that nε > x for any real number x?
A) Completeness Property
B) Order Property
C) Archimedean Property
D) Density Property
48. Consider the set A = {x ∈ R | x < 5}. Which of the following is the least upper bound of A?
A) 4
B) 5
C) 6
D) Does not exist
49. If a set S of real numbers is bounded above, its least upper bound (supremum) must be:
A) Less than or equal to every element in S
B) Greater than or equal to every element in S
C) Less than or equal to every upper bound of S
D) Greater than or equal to every upper bound of S
50. Which property of real numbers states that for any non-empty set of real numbers that is bounded above, there exists a least upper bound in the set of real numbers?
A) Archimedean Property
B) Completeness Property
C) Trichotomy Property
D) Transitive Property
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