The Picard-Lindelof Theorem is not quite correctly stated in your question. Recall that it is usually referred to as the LOCAL existence and uniqueness theorem, and it only guarantees a solution on a certain maximal interval [0,T). It 0,T), and for as simple a system as $x'=x^2$ the maximal existence time T is finite. That said, it is correct (with caveats) that the Euler method approximating solutions will converge to this "true" solution at all points of this maximal interval. The detailed story, with all the error estimates, is too complicated to state here, but you can find a careful discussion of this not only for Euler's Method, but also for a number of other methods, in Chapter 5 (Numerical Methods) of the book Differential Equations, Mechanics, and Computation (which I wrote together with my son Bob). There is a website for the book at http://ode-math.com where you can download for free more than half the book. In particular clicking here:
will download the first 38 pages of chapter 5, where starting on page 144 you will find a careful discussion of the rate of convergence and stability properties etc, for Euler's method, starting from scratch.