Here are a few analysis problems I am having trouble with:

1. Let {a_n} be a sequence of real numbers and let r be a real number that satisfies 0 < r < 1. Suppose that | a_n+1 - a_n | <= r*| a_n - a_n-1| for n > 1. Prove that {a_n} is a Cauchy sequence and hence a sequence that converges to a limit.

2. Let {a_n} be a sequence of real numbers. Prove that if {a_n} has two subsequences that converge to different limits, then the sequence {a_n} does not converge.

3. Prove the following theorem: Let {a_n} be a bounded sequence of real numbers and let S be the set of sequential limits of {a_n}. Then the set S contains its greatest lower bound and its least upper bound and

lim inf a_n = inf S & lim sup a_n = sup S.

4. Let S be a bounded nonempty set of real numbers and suppose that supS is not an element of S. Prove that there is a nondecreasing sequence {s_n} of elements of S such that the limit as n approaches infinity of s_n is supS.

Thanks for the help.