Wednesday, November 4, 2009
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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