Lecture Notes: Introduction to Real Analysis (Course 18.100)

Course Introduction

• Purpose of the course:
  • Gain experience with proofs: reading and writing proofs.
  • Prove statements about real numbers, functions, and limits (analysis).

Initial Topics: Sets and Notation

• Definition (Dfn): A set is a collection of objects called elements or members.
  • Empty Set: A set with no elements, denoted by a circle with a dash.
• Notation:
  • a ∈ S: a is an element of set S.
  • a ∉ S: a is not an element of S.
  • ∀: for all.
  • ∃: there exists.
  • ⇒: implies.
  • ⇔: if and only if.
• Subsets: A ⊆ B
  • A is a subset of B if every element of A is in B.
  • Proper Subset: A ⊂ B if A ⊆ B and A ≠ B.
• Describing Sets:
  • Use braces {} and describe properties.

Basic Set Examples

• Natural Numbers (ℕ): 1, 2, 3, etc.
• Integers (ℤ): ..., -2, -1, 0, 1, 2, etc.
• Rational Numbers (ℚ): Fractions m/n where m, n ∈ ℤ, n ≠ 0.
• Real Numbers (ℝ): Will be properly defined later in the course.

Set Operations

• Union (A ∪ B): All elements in A or B.
• Intersection (A ∩ B): Elements common to both A and B.
• Set Difference (A \ B): Elements in A not in B.
• Complement (A^c): Elements not in A.
• Disjoint Sets: A ∩ B = Ø (empty set).

Theorem: De Morgan's Laws

• Statements:
  • (B ∪ C)^c = B^c ∩ C^c
  • (B ∩ C)^c = B^c ∪ C^c
• Proof Technique: Logic and Definitions
  • Use subset definitions and logical implications.

Proofs and Logic

• Structure of a Proof:
  • Start with hypotheses.
  • Use logic and calculations to arrive at conclusions.
  • End with a conclusion (often marked by a box).

Induction

• Principle of Mathematical Induction:
  • If P(1) is true (base case) and P(n) ⇒ P(n+1) (inductive step), then P(n) is true for all natural numbers.
• Well-Ordering Property: Every non-empty subset of natural numbers has a least element.

Practice with Induction

• Example: Prove summation formula for geometric series using induction.
• Steps:
  • Verify base case (P(1)).
  • Prove inductive step, assuming P(n) to show P(n+1).

Additional Example

• Inequality with Induction:
  • Given c ≥ -1, show 1 + c^n ≥ 1 + n*c for all n.
  • Use algebraic manipulation and inductive reasoning.*

Conclusion

• Key Skills:
  • Understanding definitions and logic.
  • Writing and understanding proofs.
  • Application of induction to prove statements.