Lecture 11: Extreme and Intermediate Value Theorem; Metric Spaces

Using sequences and properties of sequences, we show both the extreme value theorem as well as the intermediate value theorem. We also define the notion of a metric space. A metric space is a space with a well-defined way of measuring distances between pairs of points. Many of the notions that we have introduced for the real numbers have counterparts for metric spaces and we discuss some of those including convergence and Cauchy sequence. We will also give a number of examples of metric spaces.