MIT's 18.100B Real Analysis, taught by Tobias Holck Colding, continues with two cornerstone results proved using sequences: the extreme value theorem, which guarantees that a continuous function on a closed bounded interval attains a maximum and minimum, and the intermediate value theorem, which guarantees it hits every value in between. Colding builds both proofs from properties of sequences developed earlier in the course rather than treating them as isolated facts. The lecture then introduces metric spaces, general sets equipped with a distance function, and shows how familiar notions from the real numbers, including convergence and Cauchy sequences, carry over once a metric is defined. Several concrete examples of metric spaces are worked through to ground the abstraction. This is lecture eleven of the term, delivered chalkboard-style in a standard MIT classroom setting, aimed at students who have already covered sequences and continuity on the real line.