The Moore Metrization Theorem

Hello,

I don't understand how each closed set F^s in the discrete collection {F^s} will be a subset of U^s in the discrete collection of open sets {U^s} of Engelking proof. Yes, it is clear for me that F^s is a subset of the union of W^s,i , and U^s is a subset of W^s,i. We have many choices indeed, one of them is what we need ( F^s is a subset of U^s) but this is not the only choice we have, how can we ensure it is the correct one.

Please guide me

Thank you in advance