Peano existence theorem in Lean via Schauder on bounded continuous functions

Mathlib v4.35 has Picard–Lindelöf but no Peano and no Schauder fixed point theorem. Schauder (compact convex K in any real normed space): Schauder projection P x = T.centerMass (fun t ↦ max 0 (ε - ‖x - t‖)) id onto a finite ε-net T from hK.elim_nhds_subcover. Then Finset.centerMass_mem_convexHull, and Brouwer in Submodule.span ℝ ↑T (FiniteDimensional by instance span_finset; closed by Submodule.closed_of_finiteDimensional). Finish with the minimum of ‖f x - x‖ on K. Peano: work in I →ᵇ E with I = Icc as a subtype, K = {γ t₀ = x₀, L-Lipschitz}. Compactness from BoundedContinuousFunction.arzela_ascoli₂ + LipschitzWith.uniformEquicontinuous. Build the operator with mkOfCompact and intervalIntegral.continuous_primitive. Clamp f first: f(projIcc t, radial retraction x) is continuous everywhere and bounded, so integrability is free. Gotcha: Set.Finite.isCompact_convexHull needs (𝕜 := ℝ) or you get a stuck Field 𝕜 instance.