asymptotic behaviour of a product

Question

Suppose $a_j\downarrow 0,b_j\downarrow 0$, and $a_j/b_j\rightarrow 1$. Do we always have $\prod_{j=1}^n\frac{1-a_j}{1-b_j}\rightarrow c$ as $n\rightarrow\infty$ for some finite constant $c$?

Thanks!

Answer 1

Here is a counter-example:

Let $a_k = \frac{1}{\sqrt{k+1}+1}$

Let $b_k = \frac{1}{\sqrt{k+1}}$

Then ( $a_k \to 0$ and $b_k \to 0$ and $\frac{a_k}{b_k} \to 1$ ) as $k \to \infty$

Also $\prod_{k=1}^n \frac{1-a_k}{1-b_k} = \prod_{k=1}^n \frac{\sqrt{k+1}^2}{\sqrt{k+1}^2-1} = \prod_{k=1}^n \frac{k+1}{k} \to \infty$ as $n \to \infty$