# Sergei Yakovenko's blog: on Math and Teaching

## Wednesday, November 4, 2009

### Lecture 1: Nov 5, 2009

Filed under: lecture,Rothschild course "Analysis for high school teachers" — Sergei Yakovenko @ 5:53

## 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
1. Sets and their subsets that have “the same number” of elements
2. Summation of infinitely many terms: success and failure
3. Limits in geometry: how to measure the lengths?
4. Pathological curves on the plane
• Numerical sequences and their limits: the first real encounter with infinity.
1. Almost all” vs. “infinitely many
2. Intervals and operations on them
3. Geometric definition of the sequence limit
4. “Instrumental” definition of the limit
5. “Standard” definition: $A=\lim_{n\to\infty}x_n \iff \forall \varepsilon\ \exists N:\ \forall n\ge N\ |x_n-A|<\varepsilon$
6. 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