# Why does this sequence fail to converge in this space?

"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|]?

• Which positive real number do you propose your sequence converge to? – user10354138 Jun 16 at 3:27

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.