Why do we not define the metric on the sequence space as the larget seperation between the sequence?

Question

Space consists of the set of all (bounded or unbounded) sequences of complex numbers and the metric $d$ defined by

$$d(x,y)=\sum_{j=1}^{\infty}\frac{1}{2^j}\frac{|\xi_j-\eta_j|}{1+|\xi_j-\eta_j|}.$$

where $x=(\xi_j)$ and $y=(\eta_j).$

I just want to ask why $d'(x,y)=\text{sup}_{j\in\Bbb{N}}|x_j-y_j|$ is not suitable for the above space?

What can I think that if we take unbounded sequence then the sup would be infinity. But I am unable to take down it in an appropriate way.

Did you mean to write$$d(x,y)=\sum_{j=1}^{\infty}\frac1{2^{\color{red}j}}\frac{|\xi_j-\eta_j|}{1+|\xi_j-\eta_j|}?$$ — José Carlos Santos, Feb 20 '21 at 16:07
Just take one of the sequences to be unbounded, and the other to be identically $0$. — saulspatz, Feb 20 '21 at 16:07

score 0 · Accepted Answer · answered Feb 21 '21 at 10:48

I just lost in the storm of overthinking...although with the help [email protected] it's clear.

If $(x_n)$ is unbounded sequence and $(y_n)$ is identically zero, then

$d'(x,y)=\text{sup}_{n\in\Bbb{N}}|x_n-y_n|=\text{sup}_{n\in\Bbb{N}}|x_n|=\infty$,

which is not a real number therefore $d'$ is not well defined and hence not suitable for said metric space.

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