Question

4.1. Let S, TCR. Prove that (i). SCT S'CT. 4.2. Let S, TC R. Prove that (i. SCT SCT. 4.3. Show that if the set S is bounded above or below, then so is S' with the bounds of S and has greatest or smallest member accordingly, provided S' 0.

Answer 1

This is a real analysis problem. We have used the definition and properties of derived set and closure of a subset of real numbers to solve the problem.
"(i) S, T Subsetequalto R S Subsetequalto T. To prove, S' Subsetequalto T' If S' = Phi, then, Phi = S' Subsetequalto T' If S' notequalto Phi, then let p elementof S' Rightarrow Forall epsilon > 0, (N_epsilon (p)\{p}) Intersection S notequalto Phi Rightarrow exist q elementof S such that, q elementof N_epsilon (p)\{p} Again, q elementof S Rightarrow q elementof T(Because S Subsetequalto T) Rightarrow q elementof (N_epsilon (p)\{p}) Intersection T Rightarrow (N_epsilon (p)\{p}) Intersection T notequalto Phi Since, epsilon > 0 is arbitrary, so p elementof T' Therefore S"" Subsetequalto T' (proved) (i) S Subsetequalto T Again, T Subsetequalto T^- Rightarrow S Subsetequalto T^- But, S^- is the smallest closed set containing S and T^- is any other closed set containing S, so S^- Subsetequalto T^- (proved) \r\n If S is finite, then S' = Phi Rightarrow S' is bounded Let, S be an infinite subset of R By, Bolzano Weierstrass theorem, S' notequalto Phi Let, S be bounded above"