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I find myself using expressions like $$a_{0_2}, (a_{n_k})_{k \in \mathbb{N}}, b_{i_{j_k}}, etc.$$

I find subsubscript and more generally, $(n \cdot \text{sub})$script for $n\geq 2$ pretty ugly and also tedious to read. Are there any notational alternatives? I'm reluctant to use superscript because of potential confusion with exponents. I sometimes use underbraces like $\underbrace{a_0}_{a_0 \in X}$, but this is only appropriate in certain contexts.

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Usually, when you end up with such towers of subindices (say, when taking subequences of subsequences of subsequences) you can rephrase the whole argument to avoid it. That is much better than changing the notation. –  Mariano Suárez-Alvarez Nov 18 '13 at 5:34
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That's the reason why math book always write "by passing to subsequence if necessary, we can assume bla bla bla". –  John Nov 18 '13 at 5:42

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You can keep track of whether you are referring to the original sequence, a subsequence, a sub-subsequence, etc., without using hard-to-read subindices, by notation such as $a_n$, $a'_n$, $a''_n$, or $a_n$, $a^{(1)}_n,$ $a^{(2)}_n,$ and so on. The suffixes here are arbitrary, of course; any distinguishing marks not already in use could be employed. This is visually unattractive, but renaming a subsequence as the original sequence can be confusing for the reader who is checking back and forth in the proof and has perhaps lost track of which sub$^k$-sequence is currently being addressed.

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