Final answer:
The set S = for all n exist in N can be the set of subsequential limits for a convergent sequence, specifically the sequence a_n = 1/n. S contains all the points to which subsequences of the given convergent sequence can converge.
Step-by-step explanation:
The question asks whether the set S = 1/n is the set of the subsequential limits of any sequence. The answer to this question is B: Yes, for a convergent sequence. A subsequential limit of a sequence is a limit of some subsequence of that sequence. In other words, for every element in the set S, we need a subsequence of the main sequence that converges to that element.
Consider the sequence an = 1/n. This sequence itself is a convergent sequence and its limit is 0, which is not in the set S. However, each element of the sequence is an element of the set S. Moreover, for every positive integer n, the constant sequence 1/n, 1/n, ... is a subsequence of an which converges to 1/n. Therefore, S can be the set of subsequential limits of this particular convergent sequence.
However, S cannot be the set of subsequential limits of any sequence, particularly not for a sequence that diverges to infinity or oscillates without converging. Hence, 'for any sequence' and 'for a divergent sequence' options are incorrect.