HW 24 extra problems

For each of the following, either give (without proof) an example of a sequence a_n that satisfies the given conditions or prove no such sequence exists.

1. lim sup a_n = 2; lim inf a_n = 1.

2. lim sup a_n = infinity; lim inf a_n = 1.

3. lim sup a_n = infinity; lim inf a_n = -infinity.

4. lim sup a_n = sup {a_n}.

5. lim sup a_n > sup {a_n}.

6. lim sup a_n < sup {a_n}.

7. lim sup a_n = 2; sup {a_n} = infinity.

8. a_n --> 2; sup {a_n} = infinity.

9. a_n --> 2; a_n is irrational for every n.

10. Every rational number is a limit point of {a_n}.

11. Every real number is a limit point of {a_n}.