Final answer:
The sequence (an) is not bounded above if and only if there exists a subsequence (ank) that goes to infinity. This means that the terms in the subsequence become larger and larger as the index, k, increases. Therefore, the correct answer is D. Goes to infinity.
Step-by-step explanation:
The sequence (an) is not bounded above if and only if there exists a subsequence (ank) that goes to infinity. This means that the terms in the subsequence become larger and larger as the index, k, increases.
To illustrate this, let's consider an example. Suppose our original sequence (an) is given by an = n, where n is a positive integer. This sequence is not bounded above because as n increases, an also increases without bound. Now, we can create a subsequence (ank) where k is even, such that ank = 2k. This subsequence goes to infinity because as k increases, ank becomes larger and larger.
Therefore, the correct answer is D. Goes to infinity.