MathDB

Define the function

f:(0,1)\to (0,1)

\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.

Let

a

and

b

be two real numbers such that

0 < a < b < 1

. We define the sequences

a_n

and

b_n

a_0 = a, b_0 = b

, and

a_n = f( a_{n -1})

b_n = f (b_{n -1} )

for

n > 0

. Show that there exists a positive integer

n

such that

(a_n - a_{n-1})(b_n-b_{n-1})<0.

Proposed by Denmark

A2