Real Analysis

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Set Theory

If you're not in a union, you're in none; if not in an intersection, you're not in all -- you might not be in one!

Axioms of the Reals

  • A1: + and · are closed binary operations on the reals.
  • A2: + and · are associative.
  • A3: + and · are commutative.
  • A4: Distributivity holds: .
  • A5: ∃ identities: and .
  • A6: ∃ additive inverses.
  • A7: ∃ multiplicative inverses (for ).
  • A8: ∃ non-empty subset P ∈ IR such that the following hold:
    1. a, b ∈ P → a + b ∈ P
    2. a, b ∈ P → a · b ∈ P
    3. a ∈ IR → (a ∈ P) ∨ (−a ∈ P) ∨ (a = 0)


  • A9: the reals are complete.

Exercise 9, p. 34 hint:

Now everything in the sum multiplied by in the final term can be bounded above. For example, we can always demand that , and we have an upper bound (call it ) on . So we can assert that

where is just some number, and must be chosen to be less than 1.

Now choose appropriately.


Some comments on proofs

Watch for some of these problems:

  • Assuming the theorem that you're in the process of proving.
  • Assuming that the "arbitrary" sets you're dealing with are denumerable, or even finite.
  • Forgetting to prove an "iff" proof in both directions.