Which of the following statements do not necessarily imply that (an) is divergent? (a) (an) is eventually positive and (1/an) is null (b) (an) is unbounded (c) (an) has two convergent subsequences whose limits are not equal (d) (an) has both an increasing subsequence and a decreasing subsequence (e) all statements imply that (an) is divergent

D does not implies that the sequence is divergent. All others statements do.

Step-by-step explanation:

Statement D: " has both an increasing subsequence and a decreasing subsequence" does not necessarily implies that the sequence is divergent. For example, let the sequence given by:

if n is odd

if n is even

We can see that the subsequence is a decreasing sequence   (the subsequence given by odd indexes). And the subsequence is an increasing sequence   (the subsequence given by even indexes).

However, is a convergent sequence with limit zero.

1/800 is the right answer

the answer is 1/4

step-by-step explanation: