Tobias Holck Colding proves the Picard-Lindelöf theorem, which establishes existence and uniqueness of solutions to first order ordinary differential equations, in this lecture from MIT's 18.100B Real Analysis course. Colding builds the proof on tools developed earlier in the course: metric spaces, Cauchy completeness, and the contraction mapping theorem. He constructs the solution to the ODE as a fixed point of a contracting map defined on the complete metric space of continuous functions over a compact interval, showing how abstract analysis machinery solves a concrete differential equations problem. The lecture runs 82 minutes and assumes familiarity with the course's earlier material on metric spaces and completeness. It is a clear demonstration of how real analysis underpins the rigorous treatment of ODE theory, aimed at students who have already seen the relevant definitions and want to watch them applied in full.