This is I think a very simple question about infinite sequences. I thought I knew the answer but the manipulation described below worries me.
Suppose I divide the interval $(0,\frac{1}{4})$ into infinitely many subintervals $S_n = (\frac{1}{(n+1)^2},\frac{1}{(n)^2})$, to wit: $S_1 = (\frac{1}{9},\frac{1}{4}),S_2 = (\frac{1}{16},\frac{1}{9})$, etc. Suppose there is a countably infinite subsequence $\{S_{n_i} \}$ of these intervals that interests me. I wish to segregate this subsequence by moving it to the left of the interval, so that the two countable sequences $\{S_n\} \setminus \{S_{n_i}\}$ and $\{S_{n_i}\}$ are segregated and remain in length-order, respectively. So we would have, $0,...S_{n_2},S_{n_1},0,...,S_2,S_1$.
Are any special assumptions needed to justify this manipulation (and is the situation clear)?
Thanks for any help.
Edit, example:
We have the line segment s: 0 _______ 1/4
I divide it into subintervals as described. Now suppose I want to take the subset of intervals indexed by odd n, and move them to the left of the segment. From the right at x = 1/4, I have a subsequence of intervals whose length approaches zero near (let us say) x = s, and then a new subsequence that begins at s, whose lengths approach 0 as they move towards x = 0. Does this help?