Final Examination Practice Problems for Metric Spaces and Convergence - Prof. Fabiola Manj, Exams of Advanced Calculus

Practice problems for the final examination covering topics such as metric spaces, completeness, compactness, interior of sets, convergence of sequences, and cauchy sequences. Students are encouraged to solve homework problems and focus on sections 1-5, 7-15, and section 13 in their preparation.

Typology: Exams

Pre 2010

Uploaded on 07/30/2009

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Practice problems for final examination
December 5, 2008
To be prepared for the midterm you must
1. know how solve every homework problem (this includes the ones you
turned in and the additional ones).
2. study sections 1-5, 7-15, section 13: pages 79- 84.
3. logic and sets
4. The final includes EVERYTHING.
Here are few extra problems that will help to practice more.
1. Give an example of a metric space that is not complete
2. Give an example of a metric space that is not compact (NOt included)
3. Denote by int(A) the interior of a set A. Is it true that int(A)int(B)=
int (AB)?
4. Let Sbe any set with dthe discrete metric. Show that every subset
ASis open.
5. Show that the sequence x(n)= (1/n, 1/n2) converges to (0,0) in R2.
6. Determine the limit of the following sequences in R2.
(a) ((sin(n))n/n, 1/n2)
(b) (1/n, nn)
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Practice problems for final examination

December 5, 2008

To be prepared for the midterm you must

  1. know how solve every homework problem (this includes the ones you turned in and the additional ones).
  2. study sections 1-5, 7-15, section 13: pages 79- 84.
  3. logic and sets
  4. The final includes EVERYTHING. Here are few extra problems that will help to practice more.
  5. Give an example of a metric space that is not complete
  6. Give an example of a metric space that is not compact (NOt included)
  7. Denote by int(A) the interior of a set A. Is it true that int(A)∪ int(B)= int (A∪ B)?
  8. Let S be any set with d the discrete metric. Show that every subset A ⊂ S is open.
  9. Show that the sequence x(n)^ = (1/n, 1 /n^2 ) converges to (0, 0) in R^2.
  10. Determine the limit of the following sequences in R^2. (a) ((sin(n))n/n, 1 /n^2 ) (b) (1/n, n−n)
  1. Let (x(n)) be a sequence in Rk^ such that d(x(n), 0) ≤ 3 for all n. Show that (x(n)) has a subsequence that converges.
  2. True or False. Justify (a) if liman=0 then ∑^ an coverges. (b) ∑^ an implies ∑^ |an| converges. (c) If (an) is bounded sequence of real numbers then (an) converges. (d) In R , a Cauchy sequence converges. (e) Every nonempty subset of R that is bounded has a supremum. (f) Q ∩ (0, 1) = ∅ (g) [4, 7) is compact in R.
  3. Let sn be a Cauchy sequence. Suppose that for every  > 0 there is n > 1 / such that |sn| < . Prove that limsn = 0.
  4. added problems Textbook problems 13.4,13.5,13.6.

GOOD LUCK!!