Given a sequence $a_n$, I know that if I can find a divergent subsequence of $a_n$, or two subsequences of $a_n$ that converge to different values, $a_n$ diverges, since, if I have understood correctly, a sequence $a_n$ converges to a limit $L$ if and only if every subsequence of $a_n$ converges to that value $L$.
I've been wondering if this last condition was equivalent to showing that some subsequences converge to $L$, picking the subsequences such that every element of the original sequence is in at least one of the subsequences. Is it? I would guess that the terms "partition" or "covering" fit this description.
Thanks.