I have a question about typical application of Noetherian induction in proofs as this one or this.
The common strategy in both examples as summarizable as that we have a (locally) Noetherian scheme $X$ and aim to show that $X$ has property $\mathcal{P}$. In the proofs instead one shows that there exist for every generic point $\eta_i$ of $X$ an afine open neighbourhood $W_i \subset X$ for which $\mathcal{P}$ holds and then one claims that applying Noetherian induction this suffice to show that $\mathcal{P}$ holds already for $X$.
Why this is a consequence of Noetherian induction? Noetherian induction says something like to show that property $\mathcal{P}$ holds for a scheme $X$ it suffice to show that for every closed $Y \subsetneq X$ if $\mathcal{P}$ holds for every closed $Y' \subsetneq Y$ then $\mathcal{P}$ holds for $Y$.
Why to apply Noetherian induction in this vein above it is sufficient in proofs above to show that there exist open neighbuorhoods of generic points for which $\mathcal{P}$ holds?