Is $ B = \left\{ \frac{n+1}{n} \,\middle|\, n \in \mathbb{N} \right\} $ as a subspace of $ ( 0 , + \infty)$ a closed set?

Question

Is $ B = \left\{ \frac{n+1}{n} \,\middle|\, n \in \mathbb{N} \right\} $ as a subspace of $ ( 0 , + \infty)$ a closed set? I would say it is not because its complement is a union of intervals and the set $(0,1] $ which is not open in $ ( 0 , + \infty)$ , right?

Answer 1

It is not closed. $1$ is a limit point of $B$, but not contained in $B$.

Answer 2

No, it is not a closed set. For instance, $\lim_{n\to\infty}\frac{n+1}n=1$, which doesn't belong to $B$.