Let $\displaystyle \alpha \in (0, 1]$ be a given real number and let a real sequence $\displaystyle \{a_n\}^{\infty}_{n=1}$ satisfy the inequality

$\displaystyle a_{n+1}\leq \alpha a_n + (1-\alpha)a_{n-1}$ for n>1

Prove that if $\displaystyle \{a_n\}$ is bounded, then it must be convergent.

my idea was to show that the sequence is monotonic, and then it would converge, but that inequality didnt really help me showing it