Let $\epsilon>0$.
I wonder if the following is true:
There exists $\lambda_0\in(0,1)$, such that for all $\lambda\leq\lambda_0$ and all $(a_n)\in [0,1]^\mathbb{N}$ there exists $N\in \mathbb{N}$ such that
$$\lambda\sum_{n=0}^{+\infty}(1-\lambda)^n a_n \geq \frac{1}{N+1} \sum_{n=0}^N a_n - \epsilon.$$