Convergent series in the space of bounded sequences I read this proposition that in the space $E$ of all bounded sequences with norm $N(u)=\sup\{|u_n|:n\in\mathbb{N}\}$, i.e. $||u||_\infty$.
the set of all convergent series is closed but those absolutely convergent is not.
I have trouble constructing a series absolutely convergent but the limit isn't...Anyone could help me out?
Thanks a lot~ ${}{}{}$

Take the usual example $$u = \left( 1 , -\frac 12 ,\frac 13, -\frac 14, \ldots \right)$$
and let $u_n$ be the same sequence, except at index $n$ stop doing the plus/minus and do zeros.  Then $$\|u_n - u\|_\infty \le \frac 1n,$$ yet each $u_n$ is absolutely convergent (being a finite sum) but $u$ clearly is not.
