Draw the following sets on the number line. Decide which one is an interval, and which one is not. Decide which intervals are closed, which ones are open, and which ones are nor open, neither closed.
1.144. A={1,2,3} 1.145. B ={x∈R: 2< x <6}
1.146. C={5.6} 1.147. D={x∈N: 2≤x≤6}
1.148. E={x∈R: 2≤x≤6} 1.149. F ={x∈R: 2< x≤6}
1.150. G={x∈R: 2≤x <6} 1.151. H ={x∈Q: 2≤x≤6}
Which ones of the following sets are bounded, bounded from above, bounded from below? Do they have minimal or maximal elements?
1.152. set of the prime numbers 1.153. set of the positive numbers
1.154. [−5,−2) 1.155.
1
n :n∈N+
1.156. {x∈R:x≤73} 1.157. {x∈Q:x≤73}
1.158. {x∈R:x≤√
2} 1.159. {x∈Q:x≤√ 2}
1.160. {n∈N:nis prime ∧ n+ 2 is prime}
1.161. Which of the following statements implies the other one?
P:The setAis finite (that is, the number of the elements ofAis finite).
Q:The setAis bounded.
1.162. Is there any sequence of numbersa1, a2, . . .such that the set{a1, a2, . . .}
is bounded, but the sequence has no maximal and no minimal ele-ments?
Write down with logical symbols the following statements.
1.163. The setAis bounded. 1.164. The set A is not bounded from below.
1.165. The setAhas no minimal element.
1.166. How many maxima, or upper bounds can a set have?
1.167. Which of the following statements implies the other one?
P:The setAhas a minimal element.
Q:The setAis bounded from below.
1.168. LetA∩B6=∅. What can we say about the connection of supA, supB, sup(A∪B), sup(A∩B) and sup(A\B)?
1.169. LetA= (0,1), B= [−√ 2,√
2] andC= 1
2n + 1
2m :n, m∈N+
. Find, if there exist, the supremum, the infimum, the maximum and the minimum of the previous sets.
1.170. LetAbe an arbitrary set of numbers, and B={−a:a∈A}, C =
1
a:a∈A, a6= 0
.
What is the connection between the supremum and the infimum of the sets?
Find, if there exist, the supremum, the infimum, the maximum and the minimum of the following sets.
1.171. [1,2] 1.172. (1,2) 1.173.
1
2n−1 :n∈N+
1.174. Q
1.175.
1 n+ 1
√n :n∈N+
1.176. √n
3 :n∈N+ 1.177. {x:x∈(0,1)∩Q} 1.178.
1 n+ 1
k :n, k∈N+
1.179. √
n+ 1−√
n:n, k∈N+ 1.180.
n+1
n:n∈N+
1.181. n√n
2 :n∈N+
o 1.182. √n
2n−n:n∈N
1.183. LetHbe a nonempty subset of the real numbers. Which of the following statements implies an other one?
(a) His not bounded from below. (b) H has no minimal element.
(c) ∀x∈H ∃y∈H (y < x). (d) ∀y∈H ∃x∈H (y < x).
1.184. We know thatcis an upper bound ofH. Does it imply that supH =c?
1.185. We know that there is no less upper bound ofH, thanc. Does it imply that supH =c?
1.186. LetAandB be not empty subsets of the real numbers. Prove that if
∀a∈A∃b∈B(a≤b), then supA≤supB.
1.187. Prove that any nonempty set, which is bounded from below, has an infimum.
Let x, y, A, B be arbitrary real numbers, and ε be a positive real number. Which of the following statements (P and Q) implies the other one?
1.188. P:|x−A|< ε Q:A−ε < x < A+ε
1.189. P:|x−y|<2ε Q:|x−A|< εand|y−A|< ε 1.190. P:|x|< Aand |y|< B Q:|x| − |y|< A−B
1.191. P:|x|< Aand |y|< B Q:|x|+|y|< A+B 1.192. P:|x|< Aand |y|< B Q:|x| − |y|< A+B
1.193. Show an example of a nonempty set of real numbers, which is bounded, but has no minimum.
1.194. Let us assume that the setH ⊂Ris nonempty. Which of the following statements implies the other one?
P:H has no minimum. Q:∀a∈R+ ∃b∈H b < a
Convergence of a Sequence
2.1 The sequence (an)convergesto the numberb∈Rif
∀ε >0∃n0∀n≥n0(|an−b|< ε).
We call the natural numbern0 thethreshold for the givenε.
If the sequence (an) converges to the number b, we can use the following notations:
n→∞lim an =b or liman=boran →b, ifn→ ∞oran→b.
If the sequence (an) is not convergent, we say that the sequence (an) is divergent.
2.2 We say that thelimitof the sequence (an) isinfinity, or (an) diverges to∞, if
∀P ∈R∃n0∀n≥n0(an> P).
The notations:
n→∞lim an=∞or liman=∞oran → ∞, ifn→ ∞oran→ ∞.
2.3 We say, that thelimitof the sequence (an) is-infinity, or (an) diverges to−∞, if
∀P ∈R∃n0∀n≥n0(an< P).
The notation:
n→∞lim an=−∞or liman=−∞oran→ −∞, orn→ ∞or an→ −∞.
2.1 Limit of a Sequence
Let the sequence (an) be: an= 1 + 1
√n. In the exercises the letters n and n0 denote positive integers.
2.1. Find a numbern0 such that∀n > n0 implies that (a) |an−1|<0,1 (b) |an−1|<0,01
2.2. Is there anyn0number such that∀n > n0implies|an−2|<0,001?
2.3. Is it true that
(a) ∀ε >0∃n0∀n > n0(|an−1|< ε) (b) ∃n0∀ε >0∀n > n0(|an−1|< ε) (c) ∃ε >0∃n0∀n > n0(|an−1|< ε) (d) ∃ε >0∃n0∀n > n0(|an−1|> ε) (e) ∀ε >0∃n0∀n≤n0(|an−1|< ε) (f ) ∀ε >0∃n0∀n≤n0(|an−1|> ε)
Find a threshold N from which all of the terms of one of the se-quences is greater than the terms of the other one.
2.4. an= 10n2+ 25 bn =n3
2.5. an = 4n5−3n2−7 bn= 10n+ 30 2.6. an= 3n−n2
bn = 2n+n
2.7. an = 2n+ 3n bn= 4n 2.8. an= 2n
bn =n!
2.9. an =n!
bn=nn 2.10. an=√
n+ 1−√ n bn = 1
n
2.11. an = 2n bn=n3
2.12. an= 0.999n bn = 1
n2
2.13. an = 10n bn=n!
Find a number N such that ∀n > N implies that 2.14. 1.01n>1000; 2.15. 0.9n< 1
100; 2.16. √n
2<1.01. 2.17. √n
n <1.0001.
2.18. n2>6n+ 15 2.19. n3>6n2+ 15n+ 37
2.20. n3−4n+ 2>6n2−15n+ 37 2.21. n5−4n2+ 2>6n3−15n+ 37
Show that there exists a numbern0such that for alln > n0implies 2.22. √
n+ 1−√
n <0.01 2.23. √
n+ 3−√
n <0.01 2.24. √
n+ 5−√
n+ 1<0.01 2.25. √
n2+ 5−n <0.01
Prove the following inequalities.
2.26. ∀n >10 2n> n3; 2.27. √
n≤1 + 1
√
2+. . .+ 1
√n <2√ n.
2.28. Which statement implies the other?
P:In the sequence (an) there is a smallest and a greatest term.
Q:The sequence (an) is bounded.
2.29. Is it true thatbis the limit of the sequence (an) if and only if
(a) for anyε >0 the sequencean has infinitely many terms closer to bthanε?
(b) for anyε >0 the sequenceanhas only finitely many terms at least εdistance tob?
(c) there exists ε >0 such that the sequencean has infinitely many terms closer tob thanε?
(d) there exists ε >0 such that the sequencean has infinitely many terms at least distanceεtob?
What can we say about the limit of the sequence (−an) if 2.30. lim
n→∞an=a(a∈R); 2.31. lim
n→∞an=∞;
2.32. lim
n→∞an=−∞? 2.33. an is oscillating divergent?
2.34. Which statement implies the other?
P: lim
n→∞an=∞
Q:(an) is bounded below, but isn’t bounded above.
Find the limits of the following sequences, and give a threshold depending on ε:
2.35. (−1)n
n 2.36. 1
√n
2.37. 1 +√ n n
2.38. n
n+ 1 2.39. 5n−1
7n+ 2 2.40. 2n6+ 3n5
7n6−2 2.41. n+n1
n+ 1
2.42. √
n+ 1−√ n
2.43. √
n2+ 1−n 2.44. 1
n−√ n
2.45. 1 +· · ·+n
n2 2.46. n
r 1 + 1
n−1
!
2.47. p
n2+ 1 +p
n2−1−2n 2.48. √3
n+ 2−√3 n−2
2.49. Are the following sequences convergent or divergent? Find the limits if they exist.
(a) an=
(3 ifnis even
4 ifnis odd (b) an=
(3 ifn≤100 4 ifn >100 (c) an=
(3n ifnis even
4n2 ifnis odd (d) an=
(n ifnis even 0 ifnis odd 2.50. Prove that the sequence 1
n does not converge to 7.
2.51. Prove that the sequence (−1)n1
n does not converge to 7.
2.52. Prove that the sequence (−1)n does not converge to 7.
2.53. Prove that the sequence (−1)n is divergent.
2.54. Prove that a convergent sequence always has a minimal or maximal term.
2.55. Show an example such thatan−bn →0 but an
bn 91.
2.56. Prove that if (an) is convergent, then also (|an|) is convergent. Is the reverse of the statement true?
2.57. Doesa2n→a2 imply thatan→a?
And doesa3n →a3imply that an →a?
2.58. Prove that ifan→a >0, then√
an →√ a.
Which statement implies that an→ ∞?
2.59. ∀K it is true that outside the interval (K,∞) the sequenceanhas only finitely many terms.
2.60. ∀K it is true that inside the interval (K,∞) the sequence an has in-finitely many terms.
2.61. Let’s assume that lim
n→∞an =∞. Which statements are true for this sequence? Which statements imply that lim
n→∞an=∞?
(a) The sequencean has no maximal term.
(b) The sequencean has a minimal term.
(c) Outside the interval (3,∞) the sequenceanhas only finitely many terms.
(d) ∀Kit is true that outside the interval (K,∞) the sequenceanhas only finitely many terms.
(e) Inside the interval (3,∞) the sequence an has infinitely many terms.
(f ) ∀K it is true that inside the interval (K,∞) the sequencean has infinitely many terms.
2.62. Is it true that if a sequence has a (finite or infinite) limit, then the sequence is bounded from below or above?
2.63. Which statement implies the other?
P:The sequence (an) is strictly monotonically increasing.
Q:The limit of (an) is infinity.
Can the limit of the sequence an be−∞,∞or a finite number, if 2.64. the sequence has infinitely many terms greater than 3?
2.65. the sequence has infinitely many terms smaller than 3?
2.66. the sequence has a maximal term?
2.67. the sequence has a minimal term?
2.68. the sequence has no minimal term?
2.69. the sequence has no maximal term?
2.70. Is there any oscillating divergent sequence, which is
(a) bounded? (b) not bounded?
2.71. A sequence has infinitely many positive and infinitely many negative terms. Can the sequence be convergent?
Find a threshold for the sequences with limit infinity:
2.72. n−√
n 2.73. 1 + 2 +· · ·+n
n 2.74.
√1 +√
2 +· · ·+√ n
n 2.75. n2−10n
10n+ 100 2.76. 2n
n 2.77. n!
2n
2.78. Find the limit of n2+ 1
n+ 1 −anifais an arbitrary real number.
2.79. Find the limit ofp
n2−n+ 1−anifais an arbitrary real number.
2.80. Find the limit ofp
(n+a)(n+b)−nifa, bare arbitrary real numbers.
2.81. Prove that ifan+1−an→c >0 , thenan→ ∞.
2.82. Prove that ifan>0, an+1 an
→c >1 , thenan→ ∞.
2.83. For which real numbers is it true that the sequence of its decimal num-bers is oscillating divergent?