Real Analysis Concepts

The final exam will cover all of the following topics.

The Real Numbers

10. The Natural Numbers

• Mathematical Induction
• Well-Ordering Principle

11. Ordered Fields

• Ordered Field Axioms (basic algebra with inequalities)
• Triangle Inequality
• All ordered fields contain \(\mathbb{Q}\)
• Examples: \(\mathbb{Q}\), \(\mathbb{R}\)

12. The Completeness Axiom

• Upper and Lower Bounds
• Bounded Sets
• The Completeness Axiom (aka Least Upper Bound Principle)
• Supremums and Infimums
• \(\mathbb{Q}\) is Dense in \(\mathbb{R}\)
• The Archimedean Principle

13. Topology

• Neighborhoods
• Interior Points and Boundary Points
• Open Sets and Closed Sets
• Accumulation Points and Isolated Points
• Closure of a Set
• Examples:
  • All open intervals \((a,b)\) are open.
    
  • All closed intervals \([a,b]\) are closed.
  • All singleton sets \(\{x\}\) are closed.
  • \(\mathbb{R}\) and \(\varnothing\) are both open and closed at the same time.

14. Compact Sets

• Compact Sets
• Open Covers and Subcovers
• Heine-Borel Theorem
• Balzano-Weierstrass Theorem

Sequences

16. Convergence

• \(\epsilon\)-\(M\) definition of limit
• Tail of a sequence
• Bounded sequences

17. Limit Theorems

• Addition/Subtraction/Multiplication rules

18. Monotone and Cauchy Sequences

• Monotone Convergence Theorem
• Cauchy Convergence Criteria

19. Subsequences

• Limit superior and limit inferior (limsup and liminf)
• Subsequential limits (the set of these is always a closed set)

Limits and Continuity

20. Limits of Functions

• \(\epsilon\)-\(\delta\) definition of limit
• Other characterizations of limits (sequential and topological)
• Limits rules (addition, multiplication, etc.)

21. Continuous Functions

• Definition of continuity at a point and on a set
• Other characterizations of continuity (sequential and topological)
  
• Continuity rules (addition, multiplication, and composition)
• Pre-images of open sets are open
• Pre-images of closed sets are closed

22. Properties of Continuous Functions

• Images of compact sets are compact
• Images of bounded sets are bounded
• Images of connected sets are connected
• Extreme Value Theorem (a continuous function achieves its maximum and minimum values on any compact domain)
• Intermediate Value Theorem

Differentiation

25. The Derivative

• Definition of derivative and differentiable
• Differentiable implies continuous

26. The Mean Value Theorem

• Fermat's Theorem (Max/min values of differentiable functions on open intervals occur when derivative is zero)
• Rolle's Theorem
• Mean Value Theorem

Integration

29. The Riemann Integral

• Definitions: Partitions, upper and lower sums, upper and lower integrals

30. Properties of the Riemann Integral

31. The Fundamental Theorem of Calculus

• Version 1
• The Evaluation Theorem (version 2)