# Question on constructing a convergent sequence

1. Oct 11, 2012

### 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)

2. Oct 12, 2012

### 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.