I'm working on topology again and I wanted to know if this proof for a question that I came on is correct.

The question was if the set {$\displaystyle \left{\frac{1}{i}\right}$}$\displaystyle ^n_{i=1}$, let's call it $\displaystyle S$, was closed in the Euclidean topology. From prior reading I knew it wasnt since it was missing it's limit point, but I couldn't use that here so I went with this.

Take $\displaystyle S'$ and suppose that it were open. There are smaller and smaller open intervals beside and not containing smaller and smaller rational numbers approaching zero.

Let $\displaystyle (a,b)$ be the open interval that surrounds zero. To the right of it there will be an open interval but to the left of that interval will be a point of S no matter how much you shrink $\displaystyle (a,b)$.

Therefore the set is not closed.