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