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"It might be well to point out that our definition of 'convergent sequence' depends not only on {$p_n$} but also on X; for instance, the sequence {$1/n$} converges in $\Bbb R^1$ (to $0$), but fails to converge in the set of all positive real numbers [with d(x,y)=|x-y|]."

Here why does the sequence {1/n} fail to converge in the set of all positive real numbers [with d(x,y)=|x-y|]?

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    $\begingroup$ Which positive real number do you propose your sequence converge to? $\endgroup$ – user10354138 Jun 16 at 3:27
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Zero is not a positive real number.

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The sequence $\{1/n\}$ as you say, converges to $0$ in $\mathbb{R}$. We note that if a sequence converges to a number, that number must be unique. So the sequence does not converge to any other number besides $0$. Since $0$ is not a positive real number, our sequence does not converge in the positive reals.

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