Question

Let (sn) be a sequence and S be the set of subsequential limits of (sn). Assume that a sequence (tn) such that tn exists in S and R has limit t. Then t exists in ______.

A. S.
B. R.
C. Both S and R.
D. Neither S nor R.

Answer 1

The limit t of the sequence (tn), which contains elements from the set of subsequential limits S, belongs to S since S is closed with respect to limits of sequences.

Step-by-step explanation:

If we have a sequence (sn) and the set S represents the subsequential limits of (sn), and we have a sequence (tn) where each tn belongs to set S, and this sequence (tn) has a limit t, then t will belong to the set S. This is because the subsequential limits of a sequence form a closed set, meaning that if a sequence of points within this set has a limit, this limit point is also within the set. Hence, the correct answer is A. S.