1549901369-Elements_of_Real_Analysis__Denlinger_

124 Chapter 2 • Sequences

• X2 is an upper bound for S;


• X2 '.S X1i
  1


• lx2 - xii~ 2;


• x 2 - ~ is not an upper bound for S, because, by definition of xi and x2,
  2
  1 1
  if x 2 =xi - -, then x2 - - =xi - 1, which is not an upper bound;
  2 1 2
  if x 2 = x 1 , then xi -
  2

is not an upper bound for S.

Define x

3

= { X2 - ~ if x2 - ~ is an upper bound for S,

x2 otherwise.

Then,

• x 3 is an upper bound for S;
  
  
  • X3 ~ X2;
    1
  
  
  
  


• lx3 - x2 I ~ 4;


• x 3 - ~ is not an upper bound for S (reason as we did to
  4 1
  show that x2 -
  2

is not an upper bound for S).

Define

x

4

= { X3 - ~ if X3 - ~ is an upper bound for S,

X3 otherwise.

We continue by mathematical induction. We define

{

1. 1.

Xk - k if Xk - k is an upper bound for S ,

Xk+i = 2 2

Xk otherwise.

In t his way, we arrive at a sequence {xn} 3 Yn EN,

• Xn is an upper bound for S;


• Xn+i ~ Xni
  1


• lxn+i - Xnl ~
  2
  n; (15)

• Xn - --i 2n- IS not an upper bound for S.