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[Merged by Bors] - feat(NormedSpace/Basic): add dist_smul_add_one_sub_smul_le #9982

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[Merged by Bors] - feat(NormedSpace/Basic): add dist_smul_add_one_sub_smul_le #9982

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11 changes: 11 additions & 0 deletions Mathlib/Analysis/NormedSpace/Basic.lean
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Expand Up @@ -86,6 +86,17 @@ variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace α E]

variable {F : Type*} [SeminormedAddCommGroup F] [NormedSpace α F]

theorem dist_smul_add_one_sub_smul_le [NormedSpace ℝ E] {r : ℝ} {x y : E} (h : r ∈ Icc 0 1) :
dist (r • x + (1 - r) • y) x ≤ dist y x :=
calc
dist (r • x + (1 - r) • y) x = ‖1 - r‖ * ‖x - y‖ := by
simp_rw [dist_eq_norm', ← norm_smul, sub_smul, one_smul, smul_sub, ← sub_sub, ← sub_add,
sub_right_comm]
_ = (1 - r) * dist y x := by
rw [Real.norm_eq_abs, abs_eq_self.mpr (sub_nonneg.mpr h.2), dist_eq_norm']
_ ≤ (1 - 0) * dist y x := by gcongr; exact h.1
_ = dist y x := by rw [sub_zero, one_mul]

theorem eventually_nhds_norm_smul_sub_lt (c : α) (x : E) {ε : ℝ} (h : 0 < ε) :
∀ᶠ y in 𝓝 x, ‖c • (y - x)‖ < ε :=
have : Tendsto (fun y => ‖c • (y - x)‖) (𝓝 x) (𝓝 0) :=
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