Question on constructing a convergent sequence

[ScroogeMcDuck]

Suppose for each given natural number n I have a convergent sequence (y_i^{(n)}) (in a Banach space) which has a limit I'll call y_n and suppose the sequence (y_n) converges to y.

Can I construct a sequence using elements (so not the limits themselves) of the sequences (y_i^{(n)}) which converges to y? I would say z_n = y_n^{(n)} would work, but I fail to prove this (my problem is making z_n^{(n)} arbitrarily small for all n bigger than some natural number M)

 

[haruspex]

in the reals, every convergent sequence has a monotonic subsequence converging to the same limit. But these could be increasing for some sequences and decreasing for others. So next you could argue there's either an infinite set of sequences with increasing subsequences or an infinite set with decreasing ones. Something along those lines might work for reals.