MathDB

Let

f:\mathbb{R} \to \mathbb{R}

be an increasing function satisfying the following conditions: 1.

f(x+1) = f(x) + 1

for each

x \in \mathbb{R}

, 2. there exists an integer p such that

f(f(f(O))) = p

. Prove that for every real number

x

\lim_{n\to \infty} \frac{x_n}{n} = \frac{p}{3}.

where

x_1 = x

and

x_n =f(x_{n-1})

for

n = 2, 3, \ldots

3