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Limit of sequence defined as a second order inequality recursion

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July 7, 2020
SequenceslimitConvergencereal analyssireal analysis

Problem Statement

Let be two real numbers b>a>0, b>a>0, and a sequence (xn)n≥1 \left( x_n \right)_{n\ge 1} with x2>x1>0 x_2>x_1>0 and such that axn+2+bxn≥(a+b)xn+1, ax_{n+2}+bx_n\ge (a+b)x_{n+1} , for any natural numbers n. n. Prove that lim⁡n→∞xn=∞. \lim_{n\to\infty } x_n=\infty .
Dan Popescu
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