Proximity to an arbitrary family and compact space

Question

https://en.wikipedia.org/wiki/Proximity_space

If we know that $A$ is not proximity to $A_\lambda$, $\forall\lambda\in\Lambda$. Is it necessarily true that $A$ is not proximity to $\cup_{\lambda\in\Lambda}A_\lambda$? [The proof]

I am asking about a theorem in our lecture notes that uses compactness (by turning arbitrary family to finite) to prove: In compact space, $A$ is near $B$ if and only if their closures intersect.

Given a compact Hausdorff space, there is a unique proximity whose corresponding topology is the given topology: $A$ is near $B$ if and only if their closures intersect. More generally, proximities classify the compactifications of a completely regular Hausdorff space.