The net you take the limit along

Definition 2.1 · def:classes · page 6

What is happening

The net class is G = { g ∈ C_c^∞(ℝ) : 0 ≤ g ≤ 1, g ≡ 1 on a nonempty open interval }, directed by g ⪯ g′ ⟺ {g = 1}° ⊆ {g′ = 1}°. Both functions take the value 1 on their plateau — that is what makes them cutoffs — so what orders them is the width of the plateau, never the height.

The rate is a function of d(g, O) = dist(O, ℝ ∖ {g = 1}°) alone: how far the observable sits inside the region where the cutoff is already switched fully on. Widen a plateau and that distance grows; Theorem 11.1 then says the states get closer in norm on A(O), at a rate faster than every inverse power of d.

This is a net, not a sequence, and the convergence is along the net itself — no subnet, and no space averaging. That is the difference from the 1970 construction, which produced an infinite-volume vacuum only as a weak-* limit point of space-averaged states.

Drag the two plateaux past each other: when neither contains the other the pair is simply not comparable, which is normal in a directed set and costs nothing, since some third cutoff dominates both.