Here are two slightly non-standard definitions that we discussed in class:

(i) A sequence {a_n} is eventually in a set V R if there exists an N such that a_n V for all n > N.

(ii) A sequence {a_n} is frequently in a set V R if, for every N, there exists n > N such that a_n V .

1. (a) Is the sequence {(1)ⁿ} eventually or frequently in the set {1}?
2. (b) Which definition is stronger? Does frequently imply eventually or does eventually imply frequently?
3. (c) Suppose an infinite number of terms of a sequence {a_n} are equal to 2. Is {a_n} necessarilyeventually in the interval (1.9, 2.1)? Is it frequently in (1.9, 2.1)?