Why a metric on a set can not be $\infty$? [duplicate]

I ask this question because I came acorss a problem (b) in exercise 3 in page 93 in section 13 in chapter 2 in the second edition of Elementary Analysis by Ross. The question is described below:

Let $B$ be the set of all bounded sequences $\boldsymbol{x} = (x_{1}, x_{2}, \cdots)$. Does $d^{*} (\boldsymbol{x}, \boldsymbol{y}) = \sum_{j = 1}^{\infty} |x_{j} - y_{j}|$ define a metric for $B$?

My answer is yes since it clearly satisfies the first two properties for a metric on a set and $|x_{j} - z_{j}| \le |x_{j} - y_{j}| + |y_{j} - z_{j}|$ hence $\sum_{j = 1}^{\infty}|x_{j} - z_{j}| \le \sum_{j = 1}^{\infty}|x_{j} - y_{j}| + \sum_{j = 1}^{\infty}|y_{j} - z_{j}|$ hence $d^{*}(\boldsymbol{x}, \boldsymbol{z}) \le d^{*}(\boldsymbol{x}, \boldsymbol{y}) + d^{*}(\boldsymbol{y}, \boldsymbol{z})$.

But I check the anwser, it says no! Because $d^{*}(\boldsymbol{x}, \boldsymbol{y})$ might be $\infty$ if $\boldsymbol{x} = (1, 1, 1, \cdots)$ and $\boldsymbol{y} = (0, 0, 0, \cdots)$.

But why the metric cannot be $\infty$? I check the definition in the book mentioned above and it does not mention anything about it. I search the questions related to it but nothing can be found.

Any help will be appreciated!