Flow Theorem via the Contraction Principle

We give an expository, self-contained proof of the classical existence, uniqueness, regularity andinvertibility theorems for flows of time-dependent vector fields, in the form used in flow-basedgenerative modelling. The proof organizes every conclusion around a single application of theBanach fixed point theorem to one operator, rather than assembling it from separate successiveapproximation, Gr\"onwall and implicit function arguments. It covers globally Lipschitz fields,local flows for Cʳ fields, the maximal flow, and a checkable growth condition for globalexistence. We also show how the same method handles fields singular at $t=1$ of the kindarising in flow matching, obtaining a Cʳ flow on $[0,1)$ with trajectories confined to a fixedball.