Let

$\displaystyle X:=\{{(a_n)^\infty_{n=0} : \sum^{\infty}_{n=0}|a_n|<\infty}\}$

be the space of absolutely covergent sequences.

Define the $\displaystyle l^1$ and $\displaystyle l^{\infty}$ metric on this space by

$\displaystyle d_{l^1}((a_n)^\infty_{n=0}, (b_n)^\infty_{n=0}):=\sum^{\infty}_{n=0}|a_n-b_n|$

$\displaystyle d_{l^\infty}((a_n)^\infty_{n=0}, (b_n)^\infty_{n=0}) := sup_{n\in\mathbb{N}}|a_n-b_n|$

First, I want to show that there exist sequences $\displaystyle x^{(1)}, x^{(2)}, ...$ of elements of $\displaystyle X$ which are convergent with respect to the $\displaystyle d_{l^{\infty}}$ metric but not with respect to the $\displaystyle d_{l^1}$ metric.

Second, that any sequence which converges in the $\displaystyle d_{l^1}$ metric, automatically converges in the $\displaystyle d_{l^{\infty}}$ metric.

Thanks