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fac6777 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingOddCompactness
namespace OAI
/-! # Pairing a bounded sequence with a strongly vanishing sequence -/
open Filter Topology
namespace DefocusingNLS
theorem tendsto_inner_of_bounded_null {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℝ H]
(u v : ℕ → H) (M : ℝ) (hu : ∀ n, ‖u n‖ ≤ M) (hv : Tendsto v atTop (𝓝 0)) :
Tendsto (fun n => inner ℝ (u n) (v n)) atTop (𝓝 0) := by
rw [tendsto_zero_iff_norm_tendsto_zero]
have ht : Tendsto (fun n => M * ‖v n‖) atTop (𝓝 0) := by
simpa only [norm_zero, mul_zero] using hv.norm.const_mul M
exact squeeze_zero (fun _ => norm_nonneg _)
(fun n => (norm_inner_le_norm (𝕜 := ℝ) (u n) (v n)).trans
(mul_le_mul_of_nonneg_right (hu n) (norm_nonneg _))) ht
end DefocusingNLS
end OAI
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