/-- Given an open set `s` in a finite-dimensional real normed vector space, there exists a smooth
function with values in `[0, 1]` whose support is exactly `s`. -/
theorem IsOpen.exists_smooth_support_eq {s : Set E} (hs : IsOpen s) :
    ∃ f : E → ℝ, f.support = s ∧ ContDiff ℝ ⊤ f ∧ Set.range f ⊆ Set.Icc 0 1 := by
  /- For any given point `x` in `s`, one can construct a smooth function with support in `s` and
    nonzero at `x`. By second-countability, it follows that we may cover `s` with the supports of
    countably many such functions, say `g i`.
    Then `∑ i, r i • g i` will be the desired function if `r i` is a sequence of positive numbers
    tending quickly enough to zero. Indeed, this ensures that, for any `k ≤ i`, the `k`-th
    derivative of `r i • g i` is bounded by a prescribed (summable) sequence `u i`. From this, the
    summability of the series and of its successive derivatives follows. -/
  rcases eq_empty_or_nonempty s with (rfl | h's)
  · exact
      ⟨fun _ => 0, Function.support_zero, contDiff_const, by
        simp only [range_const, singleton_subset_iff, left_mem_Icc, zero_le_one]⟩
  let ι := { f : E → ℝ // f.support ⊆ s ∧ HasCompactSupport f ∧ ContDiff ℝ ⊤ f ∧ range f ⊆ Icc 0 1 }
  obtain ⟨T, T_count, hT⟩ : ∃ T : Set ι, T.Countable ∧ ⋃ f ∈ T, support (f : E → ℝ) = s := by
    have : ⋃ f : ι, (f : E → ℝ).support = s := by
      refine' Subset.antisymm (iUnion_subset fun f => f.2.1) _
      intro x hx
      rcases exists_smooth_tsupport_subset (hs.mem_nhds hx) with ⟨f, hf⟩
      let g : ι := ⟨f, (subset_tsupport f).trans hf.1, hf.2.1, hf.2.2.1, hf.2.2.2.1⟩
      have : x ∈ support (g : E → ℝ) := by
        simp only [hf.2.2.2.2, Subtype.coe_mk, mem_support, Ne.def, one_ne_zero, not_false_iff]
      exact mem_iUnion_of_mem _ this
    simp_rw [← this]
    apply isOpen_iUnion_countable
    rintro ⟨f, hf⟩
    exact hf.2.2.1.continuous.isOpen_support
  obtain ⟨g0, hg⟩ : ∃ g0 : ℕ → ι, T = range g0 := by
    apply Countable.exists_eq_range T_count
    rcases eq_empty_or_nonempty T with (rfl | hT)
    · simp only [iUnion_false, iUnion_empty] at hT
      simp only [← hT, mem_empty_iff_false, iUnion_of_empty, iUnion_empty, Set.not_nonempty_empty]
          at h's
    · exact hT
  let g : ℕ → E → ℝ := fun n => (g0 n).1
  have g_s : ∀ n, support (g n) ⊆ s := fun n => (g0 n).2.1
  have s_g : ∀ x ∈ s, ∃ n, x ∈ support (g n) := by
    intro x hx
    rw [← hT] at hx
    obtain ⟨i, iT, hi⟩ : ∃ (i : ι) (_ : i ∈ T), x ∈ support (i : E → ℝ) := by
      simpa only [mem_iUnion] using hx
    rw [hg, mem_range] at iT
    rcases iT with ⟨n, hn⟩
    rw [← hn] at hi
    exact ⟨n, hi⟩
  have g_smooth : ∀ n, ContDiff ℝ ⊤ (g n) := fun n => (g0 n).2.2.2.1
  have g_comp_supp : ∀ n, HasCompactSupport (g n) := fun n => (g0 n).2.2.1
  have g_nonneg : ∀ n x, 0 ≤ g n x := fun n x => ((g0 n).2.2.2.2 (mem_range_self x)).1
  obtain ⟨δ, δpos, c, δc, c_lt⟩ :
    ∃ δ : ℕ → ℝ≥0, (∀ i : ℕ, 0 < δ i) ∧ ∃ c : NNReal, HasSum δ c ∧ c < 1
  exact NNReal.exists_pos_sum_of_countable one_ne_zero ℕ
  have : ∀ n : ℕ, ∃ r : ℝ, 0 < r ∧ ∀ i ≤ n, ∀ x, ‖iteratedFDeriv ℝ i (r • g n) x‖ ≤ δ n := by
    intro n
    have : ∀ i, ∃ R, ∀ x, ‖iteratedFDeriv ℝ i (fun x => g n x) x‖ ≤ R := by
      intro i
      have : BddAbove (range fun x => ‖iteratedFDeriv ℝ i (fun x : E => g n x) x‖) := by
        apply
          ((g_smooth n).continuous_iteratedFDeriv le_top).norm.bddAbove_range_of_hasCompactSupport
        apply HasCompactSupport.comp_left _ norm_zero
        apply (g_comp_supp n).iteratedFDeriv
      rcases this with ⟨R, hR⟩
      exact ⟨R, fun x => hR (mem_range_self _)⟩
    choose R hR using this
    let M := max (((Finset.range (n + 1)).image R).max' (by simp)) 1
    have δnpos : 0 < δ n := δpos n
    have IR : ∀ i ≤ n, R i ≤ M := by
      intro i hi
      refine' le_trans _ (le_max_left _ _)
      apply Finset.le_max'
      apply Finset.mem_image_of_mem
      -- Porting note: was
      -- simp only [Finset.mem_range]
      -- linarith
      simpa only [Finset.mem_range, Nat.lt_add_one_iff]
    refine' ⟨M⁻¹ * δ n, by positivity, fun i hi x => _⟩
    calc
      ‖iteratedFDeriv ℝ i ((M⁻¹ * δ n) • g n) x‖ = ‖(M⁻¹ * δ n) • iteratedFDeriv ℝ i (g n) x‖ := by
        rw [iteratedFDeriv_const_smul_apply]; exact (g_smooth n).of_le le_top
      _ = M⁻¹ * δ n * ‖iteratedFDeriv ℝ i (g n) x‖ := by
        rw [norm_smul, Real.norm_of_nonneg]; positivity
      _ ≤ M⁻¹ * δ n * M := (mul_le_mul_of_nonneg_left ((hR i x).trans (IR i hi)) (by positivity))
      _ = δ n := by field_simp
  choose r rpos hr using this
  have S : ∀ x, Summable fun n => (r n • g n) x := fun x ↦ by
    refine' .of_nnnorm_bounded _ δc.summable fun n => _
    rw [← NNReal.coe_le_coe, coe_nnnorm]
    simpa only [norm_iteratedFDeriv_zero] using hr n 0 (zero_le n) x
  refine' ⟨fun x => ∑' n, (r n • g n) x, _, _, _⟩
  · apply Subset.antisymm
    · intro x hx
      simp only [Pi.smul_apply, Algebra.id.smul_eq_mul, mem_support, Ne.def] at hx
      contrapose! hx
      have : ∀ n, g n x = 0 := by
        intro n
        contrapose! hx
        exact g_s n hx
      simp only [this, mul_zero, tsum_zero]
    · intro x hx
      obtain ⟨n, hn⟩ : ∃ n, x ∈ support (g n); exact s_g x hx
      have I : 0 < r n * g n x := mul_pos (rpos n) (lt_of_le_of_ne (g_nonneg n x) (Ne.symm hn))
      exact ne_of_gt (tsum_pos (S x) (fun i => mul_nonneg (rpos i).le (g_nonneg i x)) n I)
  · refine'
      contDiff_tsum_of_eventually (fun n => (g_smooth n).const_smul (r n))
        (fun k _ => (NNReal.hasSum_coe.2 δc).summable) _
    intro i _
    simp only [Nat.cofinite_eq_atTop, Pi.smul_apply, Algebra.id.smul_eq_mul,
      Filter.eventually_atTop, ge_iff_le]
    exact ⟨i, fun n hn x => hr _ _ hn _⟩
  · rintro - ⟨y, rfl⟩
    refine' ⟨tsum_nonneg fun n => mul_nonneg (rpos n).le (g_nonneg n y), le_trans _ c_lt.le⟩
    have A : HasSum (fun n => (δ n : ℝ)) c := NNReal.hasSum_coe.2 δc
    simp only [Pi.smul_apply, smul_eq_mul, NNReal.val_eq_coe, ← A.tsum_eq, ge_iff_le]
    

Proof state:
case inr.intro.intro.intro.intro.intro.intro.intro.refine'_3.intro
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
hs : IsOpen s
h's : Set.Nonempty s
ι : Type u_1 := { f // support f ⊆ s ∧ HasCompactSupport f ∧ ContDiff ℝ ⊤ f ∧ range f ⊆ Icc 0 1 }
T : Set ι
T_count : Set.Countable T
hT : ⋃ f ∈ T, support ↑f = s
g0 : ℕ → ι
hg : T = range g0
g : ℕ → E → ℝ := fun n => ↑(g0 n)
g_s : ∀ (n : ℕ), support (g n) ⊆ s
s_g : ∀ x ∈ s, ∃ n, x ∈ support (g n)
g_smooth : ∀ (n : ℕ), ContDiff ℝ ⊤ (g n)
g_comp_supp : ∀ (n : ℕ), HasCompactSupport (g n)
g_nonneg : ∀ (n : ℕ) (x : E), 0 ≤ g n x
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
δc : HasSum δ c
c_lt : c < 1
r : ℕ → ℝ
rpos : ∀ (n : ℕ), 0 < r n
hr : ∀ (n i : ℕ), i ≤ n → ∀ (x : E), ‖iteratedFDeriv ℝ i (r n • g n) x‖ ≤ ↑(δ n)
S : ∀ (x : E), Summable fun n => (r n • g n) x
y : E
A : HasSum (fun n => ↑(δ n)) ↑c
⊢ ∑' (n : ℕ), r n * ↑(g0 n) y ≤ ∑' (b : ℕ), ↑(δ b)

Next tactic:
apply tsum_le_tsum _ (S y) A.summable