Question

Let (sn) be any sequence in R, and let S denote the set of subsequential limits of (sn). What 3 things are true about S?

A. S is bounded.
B. S is closed.
C. S is non-empty.
D. All of the above.

Answer 1

The set of subsequential limits, S, of a sequence (sn) in R is bounded, closed, and non-empty.

Step-by-step explanation:

Three things that are true about the set of subsequential limits, S, of a sequence (sn) in R are:

1. S is bounded: This means that the set S has both an upper bound and a lower bound. In other words, there are finite values that the subsequential limits can take.
2. S is closed: This means that S contains all of its limit points. If a sequence has a limit, that limit will be in S.
3. S is non-empty: This means that there is at least one subsequential limit in S. It is possible for a sequence to have infinite subsequential limits, so there will always be at least one limit in S.