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prove periodicity of sequence

Source: 1998 France MO P2

April 10, 2021
Sequencealgebra

Problem Statement

Let (un)(u_n) be a sequence of real numbers which satisfies un+2=∣un+1∣−unfor all n∈N.u_{n+2}=|u_{n+1}|-u_n\qquad\text{for all }n\in\mathbb N.Prove that there exists a positive integer pp such that un=un+pu_n=u_{n+p} holds for all n∈Nn\in\mathbb N.
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