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Mathlib.Analysis.Hofer

Hofer's lemma #

This is an elementary lemma about complete metric spaces. It is motivated by an application to the bubbling-off analysis for holomorphic curves in symplectic topology. We are very far away from having these applications, but the proof here is a nice example of a proof needing to construct a sequence by induction in the middle of the proof.

References: #

theorem hofer {X : Type u_1} [MetricSpace X] [CompleteSpace X] (x : X) (ε : ) (ε_pos : 0 < ε) {ϕ : X} (cont : Continuous ϕ) (nonneg : ∀ (y : X), 0 ϕ y) :
ε' > 0, ∃ (x' : X), ε' ε dist x' x 2 * ε ε * ϕ x ε' * ϕ x' ∀ (y : X), dist x' y ε'ϕ y 2 * ϕ x'