Let D be the set of rational numbers that may be expressed as k/2, where k...

 

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Let D be the set of rational numbers that may be expressed as k/2", where k € Z and n N. The goal of this problem is to show that D is dense in Ri.e. D= R. Let D₁ = {k/2" | k= 0,1,2,...,2") and set D = =ÜDR. 1. Let A and B be subsets of R. Show that A C B = A SB. 2. Prove that for all n € N, D, C [0, 1]. 3. Prove that D[0, 1]. 4. Prove that D[0, 1]. 5. Let x € [0, 1]. Split the interval [0, 1] into two equal subintervals and call I₁ = [a, b] the part containing z. Repeat the split with I₁: divide I; into two equal parts and name 12 = [02, 6₂] the portion containing z. After k splits, we have r € [a, b]. a) Prove by induction that for all k N. a, b D and bak = 2-k b) Prove that for all k € N, 0≤z-a ≤2-*. c) Prove that a →→ ras k→ oo, and b→ I ask →→∞o. 6. Prove that [0, 1] D... 7. Prove that D = [0, 1]. 8. Let y € R. a) Prove that 0 ≤ y - [y] <1, where [] is the greatest integer function. b) Prove that there is a sequence {u} of D such that →y as k→ ∞o. c) Prove that D = R. Let D be the set of rational numbers that may be expressed as k/2", where k € Z and n N. The goal of this problem is to show that D is dense in Ri.e. D= R. Let D₁ = {k/2" | k= 0,1,2,...,2") and set D = =ÜDR. 1. Let A and B be subsets of R. Show that A C B = A SB. 2. Prove that for all n € N, D, C [0, 1]. 3. Prove that D[0, 1]. 4. Prove that D[0, 1]. 5. Let x € [0, 1]. Split the interval [0, 1] into two equal subintervals and call I₁ = [a, b] the part containing z. Repeat the split with I₁: divide I; into two equal parts and name 12 = [02, 6₂] the portion containing z. After k splits, we have r € [a, b]. a) Prove by induction that for all k N. a, b D and bak = 2-k b) Prove that for all k € N, 0≤z-a ≤2-*. c) Prove that a →→ ras k→ oo, and b→ I ask →→∞o. 6. Prove that [0, 1] D... 7. Prove that D = [0, 1]. 8. Let y € R. a) Prove that 0 ≤ y - [y] <1, where [] is the greatest integer function. b) Prove that there is a sequence {u} of D such that →y as k→ ∞o. c) Prove that D = R.