TITLE: Are there alternative proofs for existence/uniqueness of ODE solutions?
QUESTION [12 upvotes]: Consider the differential equation $\dot x = f(x)$. The standard proofs are 

The Picard iteration based proof of existence/uniqueness for Lipschitz $f$. 
The Peano existence theorem for continuous $f$.
The Caratheodory existence theorem for measurable $f$.

My question is as follows. Assuming a Lipschitz $f(x)$, are there any other proofs out there for existence of solutions (in some reasonable sense) for ODEs?

REPLY [4 votes]: The simplest proof (though 1 variable is probably not what you want)
is in Peter Lax's Calculus book, Springer, 1976, page 442. It uses
only the fundamental theorem of calculus. Existence, uniqueness, and 
asymptotics are proved for the case there.