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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. 1