# ULTRA: Real Analysis

- Author:
- Nick Wasserman

Supplementary GeoGebra files for a Real Analysis course, in relation to NSF-funded project ULTRA: Upgrading Learning for Teachers in Real Analysis (DUE-1524681, DUE-1524739, DUE-1524619).
The files can be used by an instructor as a supplement to a class lecture, but were designed to allow students to interact with them also (e.g., alter the givens, etc.).

## Table of Contents

### Introductory Aspects of Real Numbers

### Sequences and Series

- Sequences
- Series: Associativity?
- Examples of Convergent and Bounded
- Proof of Convergence for Sequences
- Example of Proof of Convergence for Sequences: Exact vs Approximate error-bound
- Approximating sqrt(7)
- Proof of if Convergent, then Bounded
- Proof of Algebraic (Scalar) Limit Theorem for Sequences
- Proof of Algebraic (Sum) Limit Theorem for Sequences
- Proof of Algebraic (Product) Limit Theorem for Sequences
- Proof of Algebraic (Reciprocal) Limit Theorem for Sequences
- Examples of Sequence Monotonicity
- Subsequences
- Proof of Bolzano-Weierstrass Theorem
- Cauchy Sequence

### Functional Limits and Continuity

### The Derivative

### Sequences and Series of Functions

### The Riemann Integral

- Equi-Partition (and Refinement) for Riemann Sums
- General Partition (and Refinement) for Riemann Sums
- Difficulties with "cut-reassemble" transformation - preparing for Cavalieri's Principle
- Integration and Cavalieri's Principle
- Cavalieri "segment-skewing" transformation for area-preservation
- Proof of Fundamental Theorem of Calculus (ii)
- Proof of Fundamental Theorem of Calculus (i)