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Prove that there exists a B when there exists an A

Source: Balkan MO ShortList 2008 A3

April 6, 2020

Problem Statement

Let

(a_m)

be a sequence satisfying

a_n \geq 0

n=0,1,2,\ldots

Suppose there exists

A >0

a_m - a_{m+1}

\geq A a_m ^2

for all

m \geq 0

. Prove that there exists

B>0

such that \begin{align*} a_n \le \frac{B}{n} \qquad \qquad \text{for }1 \le n \end{align*}