Here are two definitions: (i) A sequence (a_n) is eventually in a set A subset o
ID: 3187857 • Letter: H
Question
Here are two definitions:(i) A sequence (a_n) is eventually in a set A subset of R if there exists an N element of N such that a_n element of A for all n >= N.
(ii) A sequence (a_n) is frequently in a set A subset of R if, for every N element of N, there exists an n >= N such that a_n element of A.
Which definition is stronger? Does frequently imply eventually and/or does eventually imply frequently?
Is the sequence (-1)^n eventually or frequently in the set {1}?
Now suppose an infinite number of terms of a sequence (x_n) are equal to 2. Is (x_n) necessarily eventually in the interval (1, 9, 2, 1)? Is it frequently in (1, 9, 2, 1)?
Explanation / Answer
Remember that sequences are infinite, so looking at the first definition: (i) A sequence (a_n) is eventually in a set A subset of R if there exists an N element of N such that a_n element of A for all n >= N. That means that after a certain point in the sequence, all of the remaining elements of the sequence (there are infinite elements remaining) will be in the Subset A. For the second definition: (ii) A sequence (a_n) is frequently in a set A subset of R if, for every N element of N, there exists an n >= N such that a_n element of A. That means that no matter how far we go in the sequence, we can always find an element of A somewhere, but there is not guarantee that every element past that point will be confined to the subspace A. Therefore the definition for eventually is stronger, and eventually implies frequently. (-1)^n is frequently in {1} because every other element of the sequence is equal to 1, so no matter how high we make n, we will always be able to find an element that is equal to 1. However is is not eventually in {1} because every other element is equal to -1, so there is never a point at which every element is the sequence is equal to 1. The sequence with an infinite number of element equal to 2 would be frequently in the interval, because if there are an infinite number of those elements, you can never get far enough in the sequence so that there would be no more 2's. However it does not necessarily have to be eventually, The sequence 2(-1)^n would be an example of one that has an infinite number of element equal to 2 but is not eventually in that interval.
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