MathDB

Let

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

be a differentiable function such that its derivative

f'

is a continuous function. Moreover, assume that for all

x\in\mathbb{R}

0\leqslant \vert f'(x)\vert\leqslant \frac{1}{2}

Define a sequence of real numbers

\{a_n\}_{n\in\mathbb{N}}

by :

a_1=1~~\text{and}~~a_{n+1}=f(a_n)~\text{for all}~n\in\mathbb{N}

Prove that there exists a positive real number

M

such that for all

n\in\mathbb{N}

\vert a_n\vert \leqslant M

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