The definition of a Cauchy sequence uses a metric to indicate that the elements in a sequence are getting closer together. A metric space is complete if all Cauchy sequences converge. Several examples are given. The set of rational numbers with the usual Euclidean distance is not complete. Continuous functions on a compact subset of Euclidean space with the supremum distance is complete. Continuous functions on a closed interval on the real line with the integral distance is not complete.

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.

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