Exercisse 2.2 Analisis riil Bartle

Exercisse 2.2 2. If

 , show that |  |  ||  || if and only if    .

Proof:

|  |  ||  || |  |  | ||  ||  |      ||  ||||  ||      ||  ||  ||||         |||| |||      | |||    |||| |||  . Since ||   ||  , then ||| |||  . So,   ||| Given    .     implies that ab = 0 or ab > 0. Given

→

←



ab = 0 implies a = 0 or b = 0.

|  |  |  |  ||  ||  ||  ||  || for b = 0 then |  |  |  |  ||  ||  ||  ||  || for a = 0 then



ab > 0implies a > 0 and b > 0. Thus, a+b > 0. So that

4. Show that

|  |      ||  ||

|  |   if and only if         

Proof:

|   |                   ( (  )                                 .    that satisfy the following inequalities: a. ||    b. |   |  

6. Find all

Solution : a.

||                                       

      So, the solution set of the inequalities above is SS:=

{  |           }.

|   |                                 Since    , then     is unusefull boundary So,        So, the SS:= * |          +

 b.

7. Find all

   that satisfy the equation |  |  |  |  .

Solution : In this problem, we can separate it into three cases:

   ii.)       iii.)   i.)

–     (  )           ii.)For      , we get     (  )     i.) For x < -1, we get

Since this statement is false, no value of x from case (ii) that satisfies the equality. iii.) For

  , we get                

From i, ii, iii, we can conclude that the value of x that satisfies the equation are -3 or 4 and it can  be denoted by SS:=

*|⋁  +