Let $S=\{\{a_n\}_{n\in\mathbb{N}}: a_n\in\mathbb{R}, \sum|a_n|<\infty\}$. Determine if the metrics $d(\{a_n\},\{b_n\})=\sum|a_n-b_n|$ and $\rho(\{a_n\},\{b_n\})=\sup|a_n-b_n|$ are equivalent metrics (equivalent in sense of convergence of sequences).
I proved that both are metrics and are well defined. Also I proved that if $\{\{x_n\}_k\}_{k\in\mathbb{N}}$ is a sequence in $S$ such that converges in $d$ then converges in $\rho$ (this part is trivial because $\rho\leq d$). I need to determine if converse is true or false.