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Intro to Formal Methods
CS 5860 Fall 2014
Lecture 17
Mon. Nov. 3, 2014
Lecture 17
Topics in Bishop’s Real Analysis
1. Field properties hold. Proposition 2.6 p.20
2. Equality is not decidable, i.e. we can’t prove ∀x, y : R.(x = y ∨ ∼ (x = y)). Why?
3. Does R form an ordered field à la Royden? What are the order relations?
Definition 2.7 p.21: x ≡ {xn } is positive, x ∈ R+ iff ∃n : Z+ .(xn > yn )
Lecture 2.8. A real number is positive iff there is a positive integer m such that
xn ≥ m1 for all n.
See Bishop’s proof p.21.
4. We need to be careful in how we think of x ∈ R+ . We need the witness to
∃n : Z+ .(xn > yn ), so we think of the pair < x, n >.
5. Comparing numbers is delicate, we do not have ∀x, y : R.(x < y ∨ x = y ∨ x > y).
We need to use
Proposition 2.16 and its Corollary:
Cor: If x, y, z ∈ R and y < z then (x < z ∨ x > y)
Definition: x 6= y iff (x < y ∨ x > y)
where x < y iff (y − x) ∈ R+
x ≥ y iff x − y ∈ R0+
Note R0+ are the non-negative reals, xn ≥ yn for all n
Some unexpected behaviors
There is a continuous function f : [0, 1] → R such that f (0) < 0 and f (1) > 0 but we cannot
find a point z ∈ [0, 1], f (z) = 0
Definition: f is continuous on an interval [a, b] iff for each > 0, we can find δ() > 0 such
that |f (x) − f (y)| ≤ whenever |x − y| ≤ δ(). We say δ is the modulus of continuity.
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