Tobias Holck Colding continues his introduction to the real numbers in this MIT 18.100B Real Analysis lecture. He develops the completeness property of the reals through the least upper bound property for bounded subsets, working through the logic carefully on the board. The lecture also demonstrates why the rational numbers fail to be complete, motivating the need for a larger number system, the reals, to do analysis properly. Running 76 minutes, the session builds directly on the prior lecture's definitions and is aimed at students already following the course's axiomatic approach to the number line. The pace is deliberate, with full proofs rather than summaries, typical of an MIT OpenCourseWare recording of a live classroom session.