## Lecture 3, September 12.

Material covered: Rudin Pages 15-17, 24-26

- Schwarz inequality.
- Triangle inequality.
- Maps, surjectivity, injectivity, bijectivity.
- Finite sets, countable sets, uncountable sets, at-most-countable sets, infinite sets.
- Countability of the integers.
- A countable union of countable sets is countable.
- Cartesian product of two countable sets is countable.
- Countability of the rationals.
- The noncountability of the set of sequences with values in {0,1}.
- Metric spaces, definition and examples - Euclidean metric, discrete metric and supremum metric.