Finally, the general case (formula_29), can be reduced to the case of formula_1 as follows: given a bounded sequence in formula_29, the sequence of first coordinates is a bounded real sequence, hence it has a convergent subsequence. One can then extract a sub-subsequence on which the second coordinates converge, and so on, until in the end we have passed from the original sequence to a subsequence formula_5 times—which is still a subsequence of the original sequence—on which each coordinate sequence converges, hence the subsequence itself is convergent.