# Math 413 Analysis I

## Key Concepts

You should be comfortable with all of the key definitions.

You should be able to state and apply all of the key results.

You should be able to sketch the proof of the key results marked with (*).
### Chapter 1: Properties of the reals

**Key Definitions**
- least upper bounds and greatest lower bounds
- Completeness Axiom
- dense sets, nowhere dense sets

**Key Results**
- Nested Interval Property (*)
- Archimedean Property (*)
- density of rationals
- existence of square roots

### Chapter 2: Sequences

**Key Definitions**
- convergent sequences
- bounded sequences, monotone sequences
- Cauchy sequences
- limsup and liminf

**Key Results**
- Algebraic and Order Limit Theorems
- Monotone Convergence Theorem (*)
- Bolzano-Weierstrass Theorem (*)
- Cauchy Criterion (*)

### Chapter 3: Topology

**Key Definitions**
- open and closed sets
- limit points and closures
- Cantor set
- compact set
- open covers and subcovers
- connected sets

**Key Results**
- basic properties of open and closed sets (*)
- Heine-Borel Theorem (version 1) compact iff closed and bounded (*)
- Heine-Borel Theorem (version 2)
compact iff every open cover has a finite subcover
- nested intersection property for compact sets (*)

### Chapter 4: Functional Limits and Continuity

**Key Definitions**
- limits of functions
- continuity
- uniform continuity
- F_sigma and G_delta sets

**Key Results**
- sequential criterion for limits (*)
- sequential criterion for continuity (*)
- topological characterization of continuity
- preservation of compact sets (*)
- Extreme Value Theorem (*)
- continuous functions on a compact set are uniformly continuous (*)
- preservation of connected sets (*)
- Intermediate Value Theorem (*)
- Sets of Discontinuity are F_sigma

### Chapter 5: The Derivative

**Key Definitions**
- differentialbility
- derivatives

**Key Results**
- differentiable functions are continuous(*)
- rules for derivatives
- Interior Extemum Theorem(*)
- Darboux's Theorem
- Rolle's Theorem (*)
- Mean Value Theorem(*)
- Cauchy's Mean Value Theorem (not on midterm 2)
- L'Hospital's Rules (not on midterm 2)
- Tangent approximations (not on midterm 2)
- Newton's Method (not on midterm 2)
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