1549901369-Elements_of_Real_Analysis__Denlinger_

2.5 Monotone Sequences 97

Theorem 2.5.11 (A Sequence Converging to .y'(i) Let a be any positive

real number. D efine the sequence {xn} inductively by

{

x1 = any positive real ~umber}

Xn+- ·

'Vn EN, Xn+l =

2

Xn.

(7)

Then {xn} converges to a positive real number whose square is a. That is,

Xn ___, fo. Moreover, 'Vn ;:=:: 2,

(8)

Proof. (a) Let a > 0, and define {xn} inductively by the above scheme.

First, we prove that {xn} is bounded below. In fact, 'Vn EN, (xn+ 1 )^2 ;:::: a. To

prove this, let n EN. From Equation (7) we have

a

2Xn+l = Xn + -;

Xn

2XnXn+l = x; + a;

x; - 2XnXn+l + a = 0.

Consider this as a quadratic equation in the variable Xn· Since it has a real

number solution, Xn, its discriminant must be ;:=:: 0. That is,

(-2Xn+1)^2 - 4(1)(a) ;:=:: O;

4(xn+1)^2 - 4a ;:=:: O;

(xn+1)^2 ;:=:: a.

(b) Next, we show that { xn} is monotone decreasing for n ;:=:: 2. That is,
'Vn;:::: 2, Xn+i ~ Xn· To prove this, let n ;:=:: 2. Then,
a
Xn+-
Xn - Xn+l = Xn - Xn
2

1 x~ - a

----

2 Xn

;:=:: 0, since x~ ;:::: a from Part (a) above

(recall that Xn > 0).