Answer
Answer the FollowingQuestions :
Q1) a) (Archimedean property) prove. If and, then there exists an such that
(2.5 marks)
b) Prove. (1) A real numbers set in is undenumerable
(2) (Rational numbers) is ordered set (2.5 marks)
c) 1) Define a sequence. And prove a the sequence is divergence
2) Define a convergence sequence. And prove a convergence sequence has a unique limit (2.5 marks)
Q2) a) Prove. If be an ordered set and. Then
(1) If, then (2) If, , then (3) If, , then (4) If , then (5) if, then (2.5 marks)
b) Define a monotone decreasing and prove a monotone sequence is bounded if and only if it is convergence. Furthermore, if is monotone increasing and bounded, then
(2.5 marks)
c) (Principle of induction). Prove. If be a statements depending on a natural numbers and suppose that (i) (basic statement) is true (ii) (induction step) if is true, then is true. Then is true for all . (2.5 marks)
Dr: Mohammed Mohammed khalaf