A compact set K contained in $(0,∞)$ and a sequence $(x_n)_n$ in K that converges to 0.

Question

Give an example or argue that it is impossible.

I argue that it is impossible because, if $(x_n)_n$ is a sequence which converges to 0, then $(x_n)_n$ must be bounded above or below by 0. As $(x_n)_n$ is in K, then the set K must have the element 0 which is false as K is in the open interval of $(0,∞)$.

This is my argument. I am not sure if this is the right argument or the right way to express it.

Answer 1

I would argue as follows: since $K $ is compact, every sequence in $K $ has a convergent subsequence with the limit in $K$. Now, as $(x_n)$ converges to $0$, so does every subsequence of $(x_n)$. But $0\notin K $.

Answer 2

Compactness is not needed only the fact that $K$ is closed: if $(x_n)$ is a sequence in a closed set that converges to $\ell$, then $\ell$ belongs to said closed set.