As part of my attempts to study calculus I've stumbled upon a conceptual question in the area of subsequences.
Take the following two sequences:
1. An = 1, 1/2, 1/3, 1/4, ..., 1/n 2. Bn = 1/2, 1/4, ..., 1/n
And assume that Bn is a subsequence of An. To my understanding a Subsequence is defined to be a sequence such that every number in Bn appears in An, and every appearance of a such number in An is in a lower index than the next appearance.
On the other hand, a Subsequence is defined to be a function that takes natural numbers and returns a new index in the original sequence. If to schematically draw it:
N N bn ---> an ---> R
This shows us that Bn receives a neutral number and returns an index in An which later returns the value of the sequence in that same N.
What bothers me is the following:
By the last definition Bn is obviously not a subsequence as all its elements aren't a natural number in the first place. However, the mapping { 2,4,6,...,2n } are indexes and they really do point to the relevant numbers in An. If that's the case, is { 2,4,6,...,2n } the real subsequence, while { 1/2, 1/4, ..., 1/n } is some representation of that subsequence?
Thanks!