MathDB

A sequence of real numbers

a_{0},\ a_{1},\ a_{2},\dots

is defined by the formula a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for} i\geq 0; here

a_0

is an arbitrary real number,

\lfloor a_i\rfloor

denotes the greatest integer not exceeding

a_i

, and

\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor

. Prove that

a_i=a_{i+2}

for

i

sufficiently large.
Proposed by Harmel Nestra, Estionia

Problem Statement