Find the limit \begin{equation*} \lim_{n \to \infty} \frac{2^n}{n} \left[ 1 - \sum_{k = 1}^{n-1} \frac{(1 - \lambda 2^{-n})^{2^k}}{2^k} \right] \end{equation*} where $\lambda > 0$.

My guess is that the limit is equal to $\lambda$.