Limit of the infinite product

Question

How can I prove this result,

$\lim\limits_{N\rightarrow\infty} \ \prod\limits_{i = 0}^{N} \left(1-\frac{a_i}{N}\right) = e^{\lim_{N\rightarrow\infty}-\frac{1}{N}\sum_{i=0}^{N} a_i}$

for $a_i \in O(1)$.

Answer 1

$$ y=\prod\limits_{i = 0}^{N} \left(1-\frac{a_i}{N}\right) \implies \log(y)=\sum\limits_{i = 0}^{N}\log\left(1-\frac{a_i}{N}\right)$$ Since $N$ is large $$\log(y)\sim -\sum\limits_{i = 0}^{N}\frac{a_i}{N}=-\frac{1}{N}\sum\limits_{i = 0}^{N}a_i$$ $$y=e^{\log(y)}\sim e^{-\frac{1}{N}\sum\limits_{i = 0}^{N}a_i}$$