A series that converges in the sup-metric, but not in the 1-metric is for example this:
x_n=(1/n,1/n,...,1/n,0,0,...) (n times 1/n)
be the space of absolutely covergent sequences.
Define the and metric on this space by
First, I want to show that there exist sequences of elements of which are convergent with respect to the metric but not with respect to the metric.
Second, that any sequence which converges in the metric, automatically converges in the metric.