limits - Prove that $lim_{ntoinfty}left[1-prod_{i=1}^{n} (1-frac{a}{i} )right]= 1$.

Let $a \in (0,1)$.

Is $\lim_{n\to\infty}\left[1-\prod_{i=1}^{n}\left(\color{red}{1-}\frac{a}{i}\right)\right] = 1$?

I can show that
$$
\lim_{n\to\infty}\left[1-\prod_{i=1}^{n} \left(1-\frac{a}{i}\right) \right] >
\lim_{n\to\infty}\left[1-\left( 1- \frac{a}{n}\right) ^ n\right] =
1- e^{-a} >

0$$

but how to prove it equals to $1$?

Answer

Use that $\log(1-x) \leq -x$ for $x \in (0,1)$. So
$$\log \left(\prod_{i=1}^n (1-\frac a i) \right) = \sum_{i=1}^n \log(1 - \frac a i) \leq \sum_{i=1}^n - \frac a i \underset{n \to \infty}{\longrightarrow} - \infty.$$