Proof by bootstrapping

The Oxford English Dictionary defines “to bootstrap” as the following: To make use of existing resources or capabilities to raise (oneself) to a new situation or state; to modify or improve by making use of what is already present. The Picard theorem [17, p. 14, Theorem 1.17]: Theorem 1. Let M be a finite dimensional Hilbert space. Let F : M → M be locally Lipschitz. Let t0 ∈ R and let u0 ∈M . Then there exist −∞ ≤ T− < t0 < T+ ≤ +∞ such that, for I = (T−, T+), there exists a unique u : I →M satisfying u(t0) = u0 and ∂tu(t) = F (u(t)), t ∈ I. If T+ is finite then ‖u(t)‖M →∞ as t→ T+, and if T− is finite then ‖u(t)‖M → ∞ as t→ T−. Taylor’s formula: Theorem 2. If f ∈ C(Br(0)), then for all x ∈ Br(0) we have