Given the following assumptions:
$1.) \max_{k\leq n} |c_{k,n}| \longrightarrow 0$ as $n \to \infty$
$2.) \sum_{k=1}^n c_{k,n} \longrightarrow \lambda$ as $n\to \infty$
$3.) \sup_{n\in \mathbb N} \sum_{k=1}^n |c_{k,n}| < \infty$
I want to prove that
$\prod_{k=1}^n (1+c_{k,n}) \longrightarrow e^\lambda \quad\text{ as } n\to \infty.$
Attempt: The idea was to take the $\log$ and Taylor expand to order $1$, then estimate the remainder. For fixed $n,k$ we have $$\log\prod_{k=1}^n (1+c_{k,n}) = \sum_{k=1}^n \log(1+c_{n,k}) = \sum_{k=1}^n c_{n,k} + \frac{1}{2}\log''(\xi)c_{n,k}^2$$ for some suitable $\xi_{n,k}$ between 0 and $c_{n,k}$. Now the first term converges to $\lambda$ by assumption but we need to control the second term. The problem here is that I don't know how to work with the $\xi_{n,k}$ properly... Is this going in the right direction?