## Wednesday, November 4, 2009

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Lecture 1: Nov 5, 2009

## Limit: the first encounter with infinity

- Introduction and logistics
- Goals of the course
- Infinity: the name of the game in Analysis
- Some examples and counterexamples
- Sets and their subsets that have “the same number” of elements
- Summation of infinitely many terms: success and failure
- Limits in geometry: how to measure the lengths?
- Pathological curves on the plane

- Numerical sequences and their limits: the first real encounter with infinity.
- “
*Almost all*” vs. “*infinitely many*“
- Intervals and operations on them
- Geometric definition of the sequence limit
- “Instrumental” definition of the limit
- “Standard” definition:
- First theorems about limits (limits and arithmetic operations, limit and rearrangements, limit and boundedness, limits and monotonicity)

- Step back: number systems. Integer, rational and real numbers. Sealing the gaps. Completeness

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