What does "up to a subsequence" mean? English is my second language. Now I have to read papers written in English, and I can't understand the phrase. Well, I get a vague idea, but that's all.
What have I done? I Googled with "up to a subsequence" (yes, with quotes), but I still can't figure out the meaning from the results.
If context is needed, this link may help.
 A: I think it must mean that although the assertion might not be true of the sequence in question, it is true of some subsequence of it.  For an example where the phrase is clearly used this way, see this answer.
This usage is not quite the same as the one described in the Wikipedia article: any equivalence relation that considered all sequences equivalent to all their subsequences would have to consider all sequences equivalent, period, which would trivialize everything.  The usage in "up to a subsequence" seems to be a slight generalization, still allowing an object to be replaced by another, related, object in order to make a statement true, but where the relationship between the objects is not necessarily one of equivalence, but just that one can be obtained from the other by a specified operation.  I think it'd read a little more clearly as "up to passing to a subsequence", but nobody put me in charge, so, shrug.
This is all educated guesswork; I've never seen "up to a subsequence" before, although I have read a reasonable amount of analysis.  In the same kind of situation, I have frequently seen phrases such as "wlog we may assume the sequence converges" (where we're supposed to replace the given sequence with a subsequence if necessary), and the more explicit "passing to a subsequence if necessary, we may assume the sequence converges".
