Inequality from a Sequence I am working on a problem and I am lead to prove the following inequality which is true based on writing out the sequence and on the fact that it should be true based on what I am trying to prove. 
Let $s_0,s_1,s_2,....$ be a sequence of positive numbers satisfying $s_0=s_1=1$ and $s_n = s_{n-1} - \alpha s_{n-2}$ where $0<\alpha<\tfrac{1}{4}$. Prove that $2s_{n+1}\geq s_n$. 
Perhaps, the following property will help, $s_{n-1}s_{n+1}\leq s_n^2$ that you can assume. 

$t_n = (2s_n - s_{n-1})$ satisfies $t_n \geq \frac{t_{n-1}}{2}$.
