This quiz is designed to test your knowledge of concepts in sequences such as convergence, Cauchy convergence, boundedness, limsup, liminf, limits, limit points, etc.

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(Key: correct, incorrect, partially correct.)

1. In the rationals , the sequence 1, 1/2, 1/3, 1/4, 1/5, 1/6, ... is
   1. Bounded.
   2. Convergent.
   3. A Cauchy sequence.
   4. A and C.
   5. B and C.
   6. A, B, and C.
      • CORRECT.
   7. None of the above.
2. In the rationals , the sequence 1, -1, 1, -1, 1, -1, ... is
   1. Bounded.
      • CORRECT.
   2. Convergent.
   3. A Cauchy sequence.
   4. A and C.
   5. B and C.
   6. A, B, and C.
   7. None of the above.
3. In the reals , the sequence 1, -1/2, 1/3, -1/4, 1/5, -1/6, ... is
   1. Bounded.
   2. Convergent.
   3. A Cauchy sequence.
   4. A and C.
   5. B and C.
   6. A, B, and C.
      • CORRECT.
   7. None of the above.
4. In the rationals , the sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ... is
   1. Bounded.
   2. Convergent.
   3. A Cauchy sequence.
   4. A and C.
      • CORRECT.
   5. B and C.
   6. A, B, and C.
   7. None of the above.
5. In the reals , the sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ... is
   1. Bounded.
   2. Convergent.
   3. A Cauchy sequence.
   4. A and C.
   5. B and C.
   6. A, B, and C.
      • CORRECT.
   7. None of the above.
6. In the rationals , the sequence 1, -2, 3, -4, 5, -6, ...is
   1. Bounded.
   2. Convergent.
   3. A Cauchy sequence.
   4. A and C.
   5. B and C.
   6. A, B, and C.
   7. None of the above.
      • CORRECT.
7. In the rationals , the sequence 0.9, -0.99, 0.999, -0.9999, 0.99999, -0.999999, ... is
   1. Bounded.
      • CORRECT.
   2. Convergent.
   3. A Cauchy sequence.
   4. A and C.
   5. B and C.
   6. A, B, and C.
   7. None of the above.
8. The lim sup of the sequence 1.1, -1.01, 1.001, -1.0001, 1.00001, ... is
   1. 1
      • CORRECT.
   2. 1.1
   3. +1 and -1
   4. -1
   5. Does not exist
   6. 
9. The lim inf of the sequence 0, -1, 0, -2, 0, -3, 0, -4, 0, -5, ... is
   1. 
      • CORRECT.
   2. 
   3. 0
   4. -1
   5. -5
   6. Does not exist
10. The sup of the sequence 1, -1/2, 1/3, -1/4, 1/5, -1/6, ... is
    1. 1
       • CORRECT.
    2. 0
    3. -1/2
    4. 
    5. Does not exist
    6. 
11. The limit points of the sequence 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, ... are
    1. Just 0
       • CORRECT.
    2. 0 and
    3. 
    4. No limit points exist.
    5. All the points 0, 1, 2, 3, ... are limit points.
    6. All positive numbers are limit points.
12. The limit points of the sequence for are
    1. Just 1/2
    2. 0, -1/3, 2/5, -3/7, ...
    3. 0 and 1/2
    4. -1/2 and 1/2
       • CORRECT.
    5. 1 and -1
    6. 1/2 and 1
    7. No limit points exist.
13. The inf of the sequence for is
    1. 0
       • CORRECT.
    2. 1
    3. 3/2
    4. 0 and 1
    5. Does not exist
    6. 
14. The least upper bound of the set is
    1. 
       • CORRECT.
    2. 0
    3. 2
    4. 4
    5. 
    6. 
    7. Does not exist
15. The supremum of the set is
    1. 
    2. 0
    3. 2
    4. 4
    5. 
       • CORRECT.
    6. 
    7. Does not exist
       • PARTIALLY. While it is true that the supremum does not exist as a real number, it does exist in the extended real number system (which includes and ).
16. The infimum of the set is
    1. 
    2. 
       • CORRECT.
    3. 0
    4. 4
    5. 
    6. 
    7. Does not exist
17. The least upper bound of the set is
    1. 
    2. 0
    3. 2
    4. 4
    5. 
       • CORRECT.
    6. 
    7. Does not exist
       • PARTIALLY. While it is true that the least upper bound (or supremum) does not exist as a real number, it does exist in the extended real number system (which includes and ).
18. The infimum of the set is
    1. 
    2. 0
    3. 2
    4. 4
       • CORRECT.
    5. 
    6. 
    7. Does not exist
19. The infimum of the empty set is
    1. 0
    2. An arbitrary real number
    3. The empty set
    4. Positive infinity
       • CORRECT. It may seem unintuitive, but every real number is a lower bound for the empty set, and thus the greatest lower bound (infimum) is .
    5. 
    6. Does not exist
20. The supremum of the empty set is
    1. 0
    2. An arbitrary real number
    3. The empty set
    4. Negative infinity
       • CORRECT. It may seem unintuitive, but every real number is an upper bound for the empty set, and thus the least upper bound (supremum) is .
    5. 
    6. Does not exist
21. Let be a sequence of real numbers. What does it mean for to converge to a limit as ?
    1. For every , there exists an such that for all .
       • CORRECT.
    2. The sequence is decreasing, i.e. for all .
    3. For every , there exists an such that for all .
    4. For every , and every , we have for all .
    5. There exists such that for every , we have for all .
    6. There exists and there exists such that for all .
    7. For every , there exists an such that for infinitely many .
       • INCORRECT. This is what it means for to be a limit point of .
22. Let be a sequence of real numbers. What does it mean for the sequence to have as a limit point?
    1. For every , there exists an such that for all .
       • INCORRECT. This is what it means for to converge to , which is a slightly different concept.
    2. For every , there exists an and such that .
    3. For every , and every , there exists such that .
    4. There exists such that for every , there exists such that .
    5. There exists and there exists such that for all .
    6. For every and , there exists such that .
       • CORRECT.
23. Let be a sequence of real numbers. What does it mean for the sequence to be bounded?
    1. There exists an such that for all .
       • CORRECT.
    2. There exists an such that for some .
    3. For each there exists an such that .
    4. For every and every we have .
    5. There exists an such that or for each .
    6. There exists an and there exists such that .
    7. The sequence either has a lower bound , or an upper bound , or both.
       • INCORRECT. One needs _both_ an upper bound and a lower bound in order to be bounded.
24. Let be a sequence of real numbers. What does it mean for to be a Cauchy sequence?
    1. For every , there exists an such that for all .
       • CORRECT.
    2. The sequence is decreasing, i.e. for all .
    3. For every , there exists an such that for all .
    4. For every , and every , we have n,m \ge N$.
    5. There exists such that for every , we have for all .
    6. There exists and there exists such that for all .
    7. For every , there exists an such that for infinitely many .
25. Let be a subset of the reals . What does it mean when we say that is the supremum of ?
    1. is an upper bound for , and for any other upper bound of we have .
       • CORRECT.
    2. is the element in which is larger than all the others.
    3. is the lim sup of every sequence in .
    4. For every , we have .
    5. For every , we have .
       • INCORRECT. This says that x is an upper bound for , but does not say that it is the least upper bound for .
    6. is an upper bound for , and is larger than all other upper bounds for .
    7. Every number less than lies in , and every number greater than lies outside of .
26. Let be a subset of the reals . What does it mean when we say that is the infimum of ?
    1. is a lower bound for , and is larger than all other lower bounds for .
       • CORRECT.
    2. is a lower bound for , and is smaller than all other lower bounds for .
    3. is the element in which is smaller than all the others.
    4. is the lim inf of every sequence in .
    5. For every , we have .
    6. For every , we have .
       • INCORRECT. This says that is an lower bound for , but does not say that it is the greatest lower bound for .
    7. Every number greater than lies in , and every number less than lies outside of .

Score:  

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