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 $C^r$ 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 $C^r$ flow on $[0,1)$ with trajectories confined to a fixedball.