Example of sequence converging in $d_{l^\infty}$ but not in $d_{l^1}$. I'll denote by $X$ the space of real sequences $(a_n)$ such that $\sum |a_n|$ converges. Let $d_{l^1}$ be the metric
$$
d_{l^1}((a_n),(b_n))=\sum|a_n-b_n|
$$
and $d_{l^\infty}$ be the metric
$$
d_{l^\infty}((a_n),(b_n))=\sup\{|a_n-b_n|\}.
$$
It seems clear to me that if a sequence (of sequences) $(x^{(n)})$ in $X$ converges with respect to $d_{l^1}$, then it must also converge in $d_{l^\infty}$. 
However, I believe I recall that $d_{l^1}$ and $d_{l^\infty}$ are not equivalent, so I'm trying to find an example of a sequence in $X$ which converges in $d_{l^\infty}$ but does not converge in $d_{l^1}$. Does anyone have an example? Cheers.

$(1/n,1/n,\ldots,1/n,0,0,\ldots)$ with $1/n$ in $n$ spots.
