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 (A∪B)?
4. Let Sbe any set with dthe discrete metric. Show that every subset
A⊂Sis 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, n−n)
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