The closure of a set and the limit point of a set are both defined. The definition of a completion is given in terms of universal properties. It is important to note that the completion of a metric space is not just another metric space, but also includes an isometry of the original space to the completion. We sketch a proof that a completion exists and that the closure of the image of that space inside the completion using the isometry is the entire completion. One construction involves replacing points with equivalence classes of Cauchy sequences. One can use this construction to construct the real numbers from the rational numbers, which you might have done in Analysis I. That proof, however, is actually significantly more involved because the real numbers form an ordered field, which has much more structure than a metric space. One has to check that all of this structure is well-defined. Unfortunately, many references omit many of the details of this proof (the details are not too difficult, but in my opinion, these details should be worked out for those taking analysis for the first time). I may at some point post my own notes on the proof of this in the future.
This is part of a series of lectures on Mathematical Analysis II. Topics covered include continuous and differentiable multi-variable functions on Euclidean space, the chain rule, the implicit function theorem, manifolds, tangent spaces, vector fields, the degree and index of a smooth map, the Euler characteristic, metric spaces, the contraction mapping theorem, existence and uniqueness of solutions to ordinary differential equations, and integral equations.
I speak rather slowly, so you may wish to increase the speed of this video.
These videos were created during the 2017 Spring semester at the UConn CETL Lightboard Room.