Question

Let t exist in R. Then t is a subsequential limit of the sequence (an) n exists in N if and only if for all epsilon greater than 0, there exists infinitely many terms of the sequence (an) within the interval _______.

A. (t - ε, t + ε).
B. (-[infinity], t).
C. (t, [infinity]).
D. (0, ε).

Answer 1

For t to be a subsequential limit of sequence (an), there must be infinitely many terms within the interval (t - ε, t + ε) for any epsilon (ε) greater than 0, making the correct answer A.

Step-by-step explanation:

The student asked about the condition for a real number t to be a subsequential limit of a sequence (an) where n exists in natural numbers (N). For a real number t to be a subsequential limit of the sequence (an), for all epsilon (ε) greater than 0, there must exist infinitely many terms of the sequence within the interval (t - ε, t + ε). Therefore, the correct answer is A. (t - ε, t + ε).

This criterion for subsequential limits relates to the concept of a sequence converging to a limit. If the sequence has infinitely many terms within every interval around t, no matter how small, this indicates that t can be approached as closely as desired by terms of the sequence, which is the definition of a subsequential limit.