Question: 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

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

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

(ii) A sequence {an} is frequently in a set V R if, for every N, there exists n > N such that an 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 {an} are equal to 2. Is {an} necessarilyeventually in the interval (1.9, 2.1)? Is it frequently in (1.9, 2.1)?

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