Tobias Holck Colding continues MIT's 18.100B Real Analysis course with the tenth lecture in the series, closing out the proof of the exponential function's basic properties. The session centers on a result concerning the product of two series, building on prior lectures' groundwork in continuity and convergence. Colding works through the argument on the board in the standard MIT OCW format, with the rigor expected of a real analysis course aimed at undergraduates who have completed a first proof-based sequence. The eighty-three minute runtime allows the proof to unfold step by step rather than being compressed into a summary. As with other lectures in the 18.100B playlist, the material assumes familiarity with earlier topics in the course, including limits, continuity, and the construction of the exponential function from series definitions, and feeds directly into later work on differentiation.