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:

• H. Hofer and C. Viterbo, The Weinstein conjecture in the presence of holomorphic spheres

theorem hofer {X : Type u_1} [MetricSpace X] [CompleteSpace X] (x : X) (ε : ℝ) (ε_pos : 0 < ε) {ϕ : X → ℝ} (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'