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Can this sequence be bounded?

Source: German pre-TST 2005, problem 4, ISL 2004, algebra problem 2

January 19, 2005
IMO ShortlistalgebraSequencebounded

Problem Statement

Let a0a_0, a1a_1, a2a_2, ... be an infinite sequence of real numbers satisfying the equation an=∣an+1−an+2∣a_n=\left|a_{n+1}-a_{n+2}\right| for all n≥0n\geq 0, where a0a_0 and a1a_1 are two different positive reals.
Can this sequence a0a_0, a1a_1, a2a_2, ... be bounded?
Proposed by Mihai Bălună, Romania
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