Properties of real numbers - order, completeness, least upper bound property, Cauchy sequences - One Line Questions

1. 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? — (L - ε, L + ε) for some L
2. Which of the following statements about a sequence (x_n) is equivalent to (x_n) being a Cauchy sequence? — (x_n) converges.
3. What is the supremum of the set S = {1 - 1/n | n ∈ N}? — 1
4. What is the least upper bound of the set D = {x ∈ R | 0 < x < 1}? — 1
5. Consider the set G = {n / (n+1) | n ∈ N}. What is the greatest lower bound (infimum) of G? — 1/2
6. Consider the set B = {x ∈ R | x² < 2}. What is the least upper bound of B? — √2
7. Consider the set C = {1/n | n ∈ N}. What is the greatest lower bound (infimum) of C? — 0
8. 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: — ε
9. Consider the set A = {x ∈ R | x < 5}. Which of the following is the least upper bound of A? — 5
10. What is the infimum of the set E = {x ∈ R | x > 5}? — 5
11. If (x_n) and (y_n) are both Cauchy sequences, then the sequence (x_n + y_n) is: — Always a Cauchy sequence
12. If a sequence (x_n) converges to L, then it implies that (x_n) is: — Always a Cauchy sequence.
13. 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? — Completeness Property
14. Which property ensures that the union of two sets, each bounded above, is also bounded above? — Transitive Property of Inequalities
15. Which property is essential for proving that every convergent sequence of real numbers is bounded? — Cauchy Sequence Criterion
16. If sup(S) = s, then for any ε > 0, the interval (s - ε, s + ε) must contain: — At least one element of S
17. Consider the set F = {x ∈ R | x² ≥ 4}. This set is: — Bounded below but not above
18. 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? — Archimedean Property
19. 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: — Density of Rational Numbers
20. Which property ensures that for any real number x, there exists a natural number n such that n > x? — Archimedean Property
21. The completeness property of the real numbers is what distinguishes them from: — Rational numbers
22. If a sequence (x_n) converges, then the sequence (x_n / n) must: — Converge to 0
23. 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: — Bounded
24. 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: — Density of Irrational Numbers
25. What is a key characteristic of Cauchy sequences in the context of real numbers? — A sequence is Cauchy if and only if it is convergent.
26. The set of real numbers R is complete. This means: — Every non-empty subset of R that is bounded above has a least upper bound in R.
27. Let S be a non-empty subset of R. If 's' is the least upper bound of S, then which statement is always true? — For every ε > 0, there exists x ∈ S such that s - ε < x ≤ s.
28. A sequence (x_n) of real numbers is called a Cauchy sequence if: — For every ε > 0, there exists an integer N such that |x_n - x_m| < ε for all n, m > N.
29. A sequence (x_n) is Cauchy if and only if: — It converges to some real number.
30. If a sequence (x_n) is NOT a Cauchy sequence, then: — There exists an ε > 0 such that for any N, there are n, m > N with |x_n - x_m| ≥ ε.
31. If a set S of real numbers is bounded above, its least upper bound (supremum) must be: — Less than or equal to every upper bound of S
32. The least upper bound property of real numbers is also known as the: — Completeness Axiom
33. If a sequence (x_n) is a Cauchy sequence, then it must be: — Bounded
34. The property that for any a, b ∈ R, if a < b, then a + c < b + c for any c ∈ R is called the: — Addition Property of Order
35. If a sequence (x_n) is monotonic and bounded, then it: — Must be a Cauchy sequence.
36. Which statement about the set of natural numbers N is FALSE? — N has a greatest element.
37. The set of all upper bounds of a non-empty set S, which is bounded above, has: — A minimum element which is the supremum of S
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: — sup(S) ≤ u
39. If a set S is bounded above by 'u' and bounded below by 'l', then: — sup(S) ≤ u and inf(S) ≥ l
40. The completeness property of real numbers is crucial for establishing the existence of: — The limit of every convergent sequence.
41. The property that every non-empty set of real numbers bounded below has a greatest lower bound (infimum) is a consequence of: — The Completeness Property applied to the set of additive inverses
42. The Archimedean Property is fundamental for proving which concept related to real numbers? — The convergence of sequences
43. Which of the following is a direct consequence of the Archimedean Property? — For any positive real number ε, there exists a natural number n such that 1/n < ε.
44. The statement 'For any a, b ∈ R, either a < b, a = b, or a > b' is the definition of the: — Trichotomy Property
45. Which property states that if a < b and b < c, then a < c for real numbers a, b, and c? — Transitive Property
46. Which of the following is NOT a property of the order relation '<' on the set of real numbers R? — Multiplication: For any a, b, c ∈ R, if a < b and c > 0, then ac < bc, and if c < 0, then ac > bc.
47. If a sequence (x_n) is a Cauchy sequence, which of the following must be true? — lim (x_{n+1} - x_n) = 0
48. Consider the sequence x_n = 1/n for n ∈ N. Is this sequence a Cauchy sequence? — Yes, because it converges to 0.
49. Consider the sequence x_n = (-1)^n / n. Is this sequence a Cauchy sequence? — Yes, because it converges to 0.
50. Which property guarantees that the set of integers Z is not complete? — Z does not satisfy the least upper bound property for all its subsets.
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