Can a finite sequence be finite, even if it's indexed by infinite index set?
That is, if $\omega(k)$ is a finite sequence, but $k$ belongs to an infinite index set, then even though $\omega$ is supposed to converge, since it's a finite sequence, it wouldn't, because there's no such $k$ to display it's limit, since indices $k$ have infinite cardinality so there's no last element?
Or perhaps this means that $\omega$ would have infinite elements, but since some elements are repeated, it's range's cardinality would still be finite and thus it's a finite sequence?
A finite sequence is a sequence whose limit exists and it's the last member of the sequence.
Think about $$a_{n}=1^{n}$$ this sequence has one element, when n is from 0 to infinity .