Is the following number set compact?

Question

I'm trying to find if the following number set is compact?

S= $\{(-1)^n+\frac{1}{n}_{n\in \mathbb{N}}\}$

I know for it to be compact it must be closed, and bounded.

Here the number set is bounded between 1.5 and -1.

Now the part if the number set is closed, but i'm getting stuck since.

Let $X$ be a metric space and $A \subset X$. $A$ is closed in $X$ iff any sequence in $A$, which converges in $X$, converges in $A$. Now here since the number set converges towards 0, as $2n \rightarrow \infty$ Since only the even numbers converge to 0, but the number set converges towards $-1$ as 2n +1 $\rightarrow \infty$ Since the odd terms only converge to -1. I know it might not be the right idea, but any help would be appreciated.

So since it's bounded, and it converges to 0, -1, would this number set be compact?

Answer 1

Your set is not compact because it is not closed.

Note that the limit points $1$ and $-1$ do not belong to your set, therefore it is not closed.