Let $s:=\mathcal{F}(\mathbb{N})=\{x = (x_j)_{j\in \mathbb{N}}|x+j\in \mathbb{C} \ \forall j\in \mathbb{N}\}$ be the space of all scalar sequences. And $d(x,y):=\sum_{j=1}^\infty 2^{-j}\frac{|x_j-y_j|}{1+|x_j-y_j|},$ provided it is a metric.
Now I want to prove $x\rightarrow y$ in $d(\cdot,\cdot)$ is equivalent to $x\rightarrow y$ componentwise, namely $x_j\rightarrow y_j$ for all $j\in\mathbb{N}$.
'$\Rightarrow$' is easy by formulating the contraposition.
How to prove "$\Leftarrow$"? Because pointwise convergence does not imply convergence in metric, I am stuck here.
Thank you in advance.