1.

a. Give an example of a sequence(xn)that has both infinitely many dominant terms, and infinitely many non-dominant terms.

b. Given an example of a sequence (xn) with exactly 5 dominant terms.

c. Give an example of a divergent sequence(xn) that has both a decreasing subsequence and an increasing subsequence

d. Given an example of a sequence(xn)with a decreasing subsequence(xnk)k, such that none of the terms xnk is dominant.

2.

Prove the statement "every bounded sequence has a converging subsequence" by combining Theorem 39 (Monotone Convergence Theorem) with Theorem 56.

3.

For each of the following sets, determine whether it can be the set of all subsequential limits of a sequence. If yes, try to given an example of such a sequence. If no, justify.

a.{1,2,3}

b. (0,5]

c. R{π}

d.∅ (i.e., does there exist a sequence with no subsequential limits?)

e. [2,3]∪[5,6]