sequence ${1/n}$ converges in R (to 0) but fails to converge in the set of all positive real number. Why? Sequence $\{\frac{1}n\}$ converges in $\mathbb{R}$ (to $0$) but fails to converge in the set of all positive real numbers.
I don't fully understand why it fails to converge in the set of all positive reals. Is it because $0$ is not in that set?
Yes. You can also say positive real number is not complete. So even the sequence is still Cauchy (this is a property concerning the sequence itself), we don't have a limit inside the space to make it convergent.
The positive reals are the open set 
$$A = (0,\infty)$$
opens sets are never compact in $\Bbb R$, and in metric spaces this is equivalent to saying that this subset is also not sequentially compact. In fact $A$ is certainly not bounded. You happened to choose a sequence in this space that has no convergent sub-sequence. So you can look at it from a topological perspective.