How to prove the sequentially compact of a set of sequences.

Question

anyone can gives me some clues on how to solve this problem? I think there is something need to do with the 1/j. But I just have no idea about how to prove this. Any suggestions appreciated! Thanks so much.

Answer 1

The intuition here should be that $D$ does not put too many constraints on $\ell^1$ and as $\ell^1$ is not sequentially compact $D$ isn't either.

With $e_1 = (1,0,\ldots), e_2 = (0,1,\ldots)$ and so on, try looking at the sequence $v_k = \sum_{i = 1}^k \frac{e_i}{i}$. Does this have a convergent subsequence in $\ell^1$?