Incompleteness of $\ell^1$ with respect to $\sup$ norm

Question

I'm trying to make an example that shows $\ell^1$, that is the space of complex sequences that the sum of the norms of their components is finite, is not complete with respect to $\sup$ norm.

And also a sequence of continuous linear functional on this space with $\sup$ norm that their limit is not a continuous linear one.

I've tried a lot to make such examples. Is there any hint? Thank you very much.

Answer 1

Let $S_k$ be the sequence $$ 1, 1/2, 1/3, ..., 1/k, 0, 0, 0, \ldots $$ Each $S_i$ is is in $\ell^1$, but the limit is the harmonic series, which is not.