1549901369-Elements_of_Real_Analysis__Denlinger_

1.2 The Order Properties 19

(c) (a, b] = {x E F: a< x:::; b};

(d) [a, b) = {x E F: a:::; x < b};

(e) (- oo,b) = {x E F: x < b};

(f) (-oo,b] = {x E F: x:::; b};

(g) (a,+oo) = {x E F: x >a};

(h) [a,+oo) = {x E F: x 2: a};

(i) (-oo, +oo) = F.

(This could be 0.)

( This could be 0 .)

(Intervals of the form (b), (e), (g), and (i) are called open intervals.)

Proof. See Exercise 4. •

EXERCISE SET 1.2-B

l. Prove Theorem 1.2.13 (a), (d), and (e).

2. Prove Theorem 1.2.14 (a) and (c).


3. Prove Theorem 1.2.15 (c) and (d).


4. Prove that the sets described in Theorem 1.2.17 (c) and (g) are intervals.


5. Intervals: Let I denote an interval, and x EI. Prove that
   I = LJ { [y, z] : y , z E I}.


6. Prove that Vx,y E ordered field F, max{x,y} x + Y + Ix - YI and
   2
   . x +y-Ix -yl
   mm{x,y} =
   2

.

7. Prove that Vx,y E ordered field F , min{x,y} = -max{- x,-y}.


8. Prove that in any ordered field, 0 < x < y =? x < Y.
   
   
   • l+x l+y
   
   
   
   


9. Prove t hat. many ordered field, Ix +YI I I :::; --lx l IYI
   1
   
   
   • 1
   
   
   
   

+ --

1

• 1

.

l+x+y l+x l+y

10. (Project)

(a) Three elements a, b, c of a field form an arithmetic progression

if their successive differences are equal: b - a = c - b. Prove that

b = a; c. [bis called the arithmetic mean of a and c.]