"Convex" sequence converges?

Question

If I have a sequence $\{ a_n\}_{n=1}^{\infty} $ that satisfy the property that (for all $x\in [0,1]$) $x \: a_{n-1} + (1-x)\: a_{n+1}> a_n$ and $a_{n+1}< a_n$ for all $n \in \mathbb{N}$. It is true that te sequence converge? I see it graphically because is like the sequence would be "convex", but I'm not sure if it is true. Thanks

Answer 1

The sequence is decreasing but may have no lower bound, hence no limit. For example if $a_n=-\ln n$ for every $n.$