MIT's 18.100B Real Analysis course, taught by Tobias Holck Colding, reaches uniform convergence and its consequences in this lecture. Colding uses the Weierstrass M-test to prove that power series are continuous inside their radius of convergence, then shows that the space C([a,b]) of continuous functions on an interval, equipped with the supremum norm, is Cauchy complete. That completeness result sets up later work on existence and uniqueness of solutions to ordinary differential equations. The second half turns to integration and differentiation, showing precisely when and how these operations commute with uniform limits of sequences of functions. The lecture is chalkboard proof work throughout, building each result from definitions rather than citing them, aimed at students already comfortable with sequences, series, and basic topology of metric spaces.