Two metrics in the space of bounded sequences in $\mathbb{C}$.

Question

Let $X$ be the set of all bounded sequences in $\mathbb{C}$. Consider the metrics $$d_1(a_n, b_n)=\sup \lbrace |a_k-b_k|:k\in \mathbb{N} \rbrace$$ $$d_2(a_n,b_n)=\sum_{k=0}^{\infty} 2^{-k}|a_k-b_k|$$ on $X$.

Do the open sets defined by these two metrics coincide?

It's clear that for all $x_n \in X$ and for all $r>0$ there is some $r'>0$ such that $B_{d_1}(x_n, r') \subset B_{d_2}(x_n, r)$ (just take $r'=r/2$). I'm stuck in the other direction.

Thanks a lot.

Answer 1

The metrics are not equivalent: let $e_1 =(1,0,..),e_2 =(0,1,...),...$ and $x_n=0$ for all n. Then $d_2 (e_n,x)=\frac 1 {2^{n}} \to 0$ but $d_1 (e_n,x)=0$ for all n.