MathDB

Let

(a_{n})_{n\ge 0}

and

(b_{n})_{n\ge 0}

be two sequences with arbitrary real values

a_0, a_1, b_0, b_1

. For

n\ge 1

, let

a_{n+1}, b_{n+1}

be defined in this way:

a_{n+1}=\dfrac{b_{n-1}+b_{n}}{2}, b_{n+1}=\dfrac{a_{n-1}+a_{n}}{2}

Prove that for any constant

c>0

there exists a positive integer

N

s.t. for all

n>N

|a_{n}-b_{n}|<c

A6