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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!

Helen
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  • Perhaps the link provided in the close box will help you. Nonetheless it is not at all clear what you are asking. By definition, a metric on a set $X$ is a certain kind of function $d : X \times X \to [0,\infty)$, and so $d(x,y)$ is a real number, not infinity. The example you gave with $x=(1,1,1,...)$ and $y=(0,0,0,...)$ clearly violates that definition and so $d^*$ is not a metric. If you are wondering why the definition of a metric is formulated to disallow $\infty$, the link in the close box should help. But if that's not what you are asking, perhaps you can edit your post to clarify. – Lee Mosher Aug 18 '24 at 17:16
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    Please check the definition in your book again. It certainly says that a metric is a map from $X \times X$ to the non-negative reals. – Ulli Aug 18 '24 at 17:18
  • So sorry that I asked a so so stupid question and did not check the definition carefully. orz – Helen Aug 20 '24 at 11:23

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