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Bounded recurrent sequence of real numbers

Source: Romanian District Olympiad 2006, Grade 11, Problem 3

March 11, 2006
probabilityreal analysisreal analysis unsolved

Problem Statement

Let {xn}n≥0\{x_n\}_{n\geq 0} be a sequence of real numbers which satisfy (x_{n+1} - x_n)(x_{n+1}+x_n+1) \leq 0,   n\geq 0. a) Prove that the sequence is bounded; b) Is it possible that the sequence is not convergent?
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