The question came up when I was trying to prove the compactness of the stone space S(B) of a complete boolean algebra B. Using only the basic facts regarding ultrafilters and boolean algebras, I cannot seem to find an answer.
Thank you very much, in advance.
The property in title seems require compactness. However, it seems not every complete boolean algebra is compact. there is a counterexample. Let $B=\wp(\omega)$, $X=\omega$. Note that $n=\{0,1,...,n-1\}$.
Then $\sup(X)=\bigcup(X)=\omega$, but for every finite $Y\subseteq X$, $\sup(Y)$ is definitely a finite number.