I have a doubt with the proof of 1/n being neither open nor closed. I have seen various replies to similar queries; however, none of them seems to be speaking of the same reasons. So, I would like to know a conclusive proof of the assertion.
Thanks Jai
On
To simplify notation, write $X = \{ \frac{1}{n} \mid n \in \mathbb{N} \}$.
To see that $X$ isn't open, note that $1 \in X$, but $1+\varepsilon \not \in X$ for any $\varepsilon > 0$.
To see that $X$ isn't closed, you can prove that its complement isn't open. To see this, note that $0 \not \in X$, but $(-\varepsilon, \varepsilon) \cap X \ne \varnothing$ for all $\varepsilon > 0$.
There are still some details to be worked out (by you).
A set is open if all its points are interior points. But no point of $\{1/n\}$ is interior, so it's not an open set.
A set is closed if it contains all its limit points. But $0$ is a limit point of $\{1/n\}$ which is not in the set, so it's not a closed set.