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> How are the elements of this sequence related

They are not necessarily related

> And why are we only interested in the elements where the index n/m goes to infinity?

If the sequence is finite, then we don’t really care to discuss the “limit” of the sequence.

> Loosely speaking, saying that a Hilbert space is complete means that it contains all of its limits.

For a set S to not contain all of its limits means you can have an infinite sequence of points (a_n) where each a_n is in S and there is no point a in S so that the sequence is eventually as close to a as you’d like.

More formally, there does not exist a in S so that for any e > 0 we can pick an M so that m > M implies | a_m - a | < e.

You can see how “m > M” gives a formal meaning to “eventually,” and “for any e > 0 … | a_m - a | < e” gives a formal meaning to “as close to a as you’d like.”