Suppose for each given natural number n I have a convergent sequence [itex](y_i^{(n)})[/itex] (in a Banach space) which has a limit I'll call [itex]y_n[/itex] and suppose the sequence [itex](y_n)[/itex] converges to [itex]y[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

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

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# Question on constructing a convergent sequence

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