Why Does the Sequence {1/n} Converge in R1 but Not in Positive Real Numbers?

[henryN7]

on page 48 of baby Rudin, it says " the sequence {1/n} converges in R1 to 0, but fails to converge in the set of all positive real numbers [with d(x,y) = |x-y|]."

ok, I know it has something to do with 1/n going to infinity near zero, but it does that whether the metric space is R1 or just the positive reals.

So why is that quote true?

 

[awkward]

on page 48 of baby Rudin, it says " the sequence {1/n} converges in R1 to 0, but fails to converge in the set of all positive real numbers [with d(x,y) = |x-y|]."

ok, I know it has something to do with 1/n going to infinity near zero, but it does that whether the metric space is R1 or just the positive reals.

So why is that quote true?

Because the positive reals do not include zero, the limit of the sequence.

 

[henryN7]

Because the positive reals do not include zero, the limit of the sequence.

Ah, thank you.