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convex-type sequence is bounded, hence convergent

Source: VJIMC 1997 1.2

September 26, 2021
Sequenceslimitsreal analysis

Problem Statement

Let α∈(0,1]\alpha\in(0,1] be a given real number and let a real sequence {an}n=1∞\{a_n\}^\infty_{n=1} satisfy the inequality an+1≤αan+(1−α)an−1for n=2,3,…a_{n+1}\le\alpha a_n+(1-\alpha)a_{n-1}\qquad\text{for }n=2,3,\ldotsProve that if {an}\{a_n\} is bounded, then it must be convergent.
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