Just finished the proof of Picard’s theorem on existence and uniqueness of solutions to ODEs in my real analysis class:
Theorem (Picard). Let be closed bounded intervals and let and be their interiors. Suppose is continuous and Lipschitz in the second variable, that is, there exists a number L such that
for all , .
Let . Then there exists an and a unique differentiable , such that
That was the last lecture for the semester, yay! The proof is essentially the standard one, but of course I don’t use the fixed point theorem on metric spaces since I don’t have that. Convergence and uniqueness is shown purely by methods taught in the class. The only thing which I had to define to state and prove the theorem was continuity in two variables. I think it’s the longest proof in the book, being 2.5 pages long (it’s 12pt font, plus it’s given in a lot of detail).
See the last section in my real analysis notes/book.
I was thinking of using something else such as implicit function theorem or some such, but Picard’s theorem is just cool. Plus the proof uses everything we’ve learned: continuity, derivatives, Riemann integral, uniform convergence, swapping of limits, etc… Plus the proof is a bunch of estimates. It doesn’t get any more “analysis” than that.