Plain-English statement
For each fixed point, the radial cutoff converges to one as its scale tends to positive infinity.
Mathematical statement
For fixed x, Tendsto (R -> chi_R(x)) atTop (nhds 1).
Intuition
Eventually R exceeds norm(x), so the scaled point lies in the unit plateau where the cutoff is exactly one.
Conditions
- The point x is fixed while R tends to infinity.
- The ambient space is normed.
Why these conditions cannot be dropped
- Pointwise convergence does not imply dominated convergence of integrals.
- Uniform derivative or tail estimates require additional independent lemmas.
Proof route
- Use the eventual inequality norm(x) <= R at atTop.
- Rewrite the cutoff by its plateau theorem.
- Conclude eventual equality and hence Tendsto.
Lean interface notes
- Tendsto is filter-level convergence.
- This leaf cannot by itself justify passing a limit through an integral.
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Lean statement
theorem radialSmoothCutoff_tendsto_one (x : E) :
Tendsto (fun R : ℝ => radialSmoothCutoff R x) atTop (𝓝 1) := by
apply tendsto_atTop_of_eventually_const (i₀ := ‖x‖ + 1)
intro R hR
have hR_pos : 0 < R := by
calc
0 < ‖x‖ + 1 := by positivity
_ ≤ R := hR
apply radialSmoothCutoff_eq_one_of_norm_le hR_pos
linarith
end Radial
section Plateau
/-- A compact subset of an open set admits a smooth compactly supported plateau in that set.
The function takes values in `[0, 1]` and is identically one on the compact set. -/Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:390published source at 7bcd37294df1