Question about the proof of "If K is a compact set of the metric space Ω, then K is closed"

I am reading the proof of the Theorem: If $K$ is a compact set of the metric space $\Omega$, then $K$ is closed, and I encounter a problem.Here is the proof in the book

What I don't understand is the last sentence. Why when $K$ belongs to $G_s$ for some $s$, and it follows that $B_{1/s}(x)$ belongs to $K$'s complement.

Thank you for answering my question.