| import Mathlib |
|
|
| namespace OAI |
|
|
| section |
| open Filter Finset |
| open scoped Topology |
|
|
| namespace InfiniteMatroidCounterexample |
|
|
| /-- An elementary weighted Cesàro estimate. -/ |
| lemma weighted_mean_zero (w d : ℕ → ℝ) (hw : ∀ n, 0 ≤ w n) |
| (hc : Tendsto (fun n ↦ ∑ m ∈ range n, w m) atTop atTop) |
| (hd : Tendsto d atTop (𝓝 0)) : |
| Tendsto (fun n ↦ (∑ m ∈ range n, w m * d m) / (∑ m ∈ range n, w m)) atTop (𝓝 0) := by |
| apply Metric.tendsto_atTop.mpr |
| intro ε hε |
| obtain ⟨N, hN⟩ := Metric.tendsto_atTop.mp hd (ε/2) (by linarith) |
| let C := ∑ m ∈ range N, |w m * d m| |
| have hC : 0 ≤ C := Finset.sum_nonneg (fun _ _ ↦ abs_nonneg _) |
| have hdiv : Tendsto (fun n ↦ C / (∑ m ∈ range n, w m)) atTop (𝓝 0) := |
| tendsto_const_nhds.div_atTop hc |
| obtain ⟨M, hM⟩ := Metric.tendsto_atTop.mp hdiv (ε/2) (by linarith) |
| obtain ⟨L, hL⟩ := eventually_atTop.mp (hc.eventually (eventually_gt_atTop (0:ℝ))) |
| refine ⟨max N (max M L), ?_⟩ |
| intro n hn |
| have hnN : N ≤ n := le_trans (le_max_left _ _) hn |
| have hnM : M ≤ n := le_trans (le_trans (le_max_left _ _) (le_max_right _ _)) hn |
| have hnL : L ≤ n := le_trans (le_trans (le_max_right _ _) (le_max_right _ _)) hn |
| have hpos := hL n hnL |
| have hb : (∑ m ∈ range n, |w m * d m|) ≤ C + ε/2 * (∑ m ∈ range n, w m) := by |
| obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_le hnN |
| rw [Finset.sum_range_add] |
| change C + _ ≤ _ |
| apply add_le_add_right |
| calc |
| ∑ m ∈ range k, |w (N+m) * d (N+m)| ≤ ∑ m ∈ range k, ε/2 * w (N+m) := by |
| apply Finset.sum_le_sum |
| intro m hm |
| rw [abs_mul, abs_of_nonneg (hw _), mul_comm] |
| apply mul_le_mul_of_nonneg_right _ (hw _) |
| have h := hN (N+m) (by omega) |
| simpa only [Real.dist_eq, sub_zero] using h.le |
| _ = ε/2 * (∑ m ∈ range k, w (N+m)) := (Finset.mul_sum _ _ _).symm |
| _ ≤ ε/2 * (∑ m ∈ range (N+k), w m) := by |
| apply mul_le_mul_of_nonneg_left _ (by linarith) |
| rw [Finset.sum_range_add] |
| have := Finset.sum_nonneg (s := range N) (fun m _ ↦ hw m) |
| linarith |
| rw [Real.dist_eq, sub_zero, abs_div, abs_of_pos hpos] |
| calc |
| |∑ m ∈ range n, w m * d m| / (∑ m ∈ range n, w m) ≤ |
| (C + ε/2 * (∑ m ∈ range n, w m)) / (∑ m ∈ range n, w m) := by |
| exact div_le_div_of_nonneg_right ((Finset.abs_sum_le_sum_abs _ _).trans hb) hpos.le |
| _ = C / (∑ m ∈ range n, w m) + ε/2 := by field_simp |
| _ < ε := by |
| have h := hM n hnM |
| rw [Real.dist_eq, sub_zero, abs_of_nonneg (div_nonneg hC hpos.le)] at h |
| linarith |
|
|
| /-- A tail supremum of absolute values of a null sequence is again null. -/ |
| lemma tail_absSup_zero (d : ℕ → ℝ) (hb : BddAbove (Set.range (fun n ↦ |d n|))) |
| (hd : Tendsto d atTop (𝓝 0)) : |
| Tendsto (fun n ↦ sSup ((fun m ↦ |d m|) '' Set.Ici n)) atTop (𝓝 0) := by |
| apply Metric.tendsto_atTop.mpr |
| intro ε hε |
| obtain ⟨N, hN⟩ := Metric.tendsto_atTop.mp hd (ε/2) (by linarith) |
| refine ⟨N, fun n hn ↦ ?_⟩ |
| have hne : ((fun m ↦ |d m|) '' Set.Ici n).Nonempty := ⟨|d n|, Set.mem_image_of_mem _ (show n ∈ Set.Ici n from (by simp))⟩ |
| have hb' : BddAbove ((fun m ↦ |d m|) '' Set.Ici n) := hb.mono (Set.image_subset_range ..) |
| have hlo : 0 ≤ sSup ((fun m ↦ |d m|) '' Set.Ici n) := |
| (abs_nonneg (d n)).trans (le_csSup hb' (Set.mem_image_of_mem _ (show n ∈ Set.Ici n from (by simp)))) |
| have hhi : sSup ((fun m ↦ |d m|) '' Set.Ici n) ≤ ε/2 := by |
| apply csSup_le hne |
| rintro _ ⟨m, hm, rfl⟩ |
| have h := hN m (hn.trans hm) |
| simpa only [Real.dist_eq, sub_zero] using h.le |
| rw [Real.dist_eq, sub_zero, abs_of_nonneg hlo] |
| linarith |
|
|
| end InfiniteMatroidCounterexample |
| end |
|
|
| end OAI |
|
|
