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Bounded sequence is constant

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October 31, 2015
algebra

Problem Statement

Suppose a doubly infinite sequence of real numbers ...,a−2,a−1,a0,a1,a2,.... . . , a_{-2}, a_{-1}, a_0, a_1, a_2, . . . has the property that an+3=an+an+1+an+23,a_{n+3} =\frac{a_n + a_{n+1} + a_{n+2}}{3}, for all integers n.n . Show that if this sequence is bounded (i.e., if there exists a number RR such that ∣an∣≤R|a_n| \leq R for all nn), then ana_n has the same value for all n.n.
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