approximation to "generalized binomial coefficient" What is the limit, when $n$ goes to $\infty$, of the following product, when $0 \leq a \leq 1$?
$$ {{1-a} \over 1}\cdot {{2-a} \over 2} \cdot {{3-a} \over 3} \cdot\ldots\cdot {{n-a} \over n} $$
When $a=0$, the product is 1, and when $a=1$, the product is 0, so I assume the product decreases monotonically with $a$ (actually, from the first factor it is clear that the product is always at most $1-a$).  But I could not find any better approximations.

We have:
$$\exp x \geq 1 + x $$ 
for all $x$. 
So:
$$ 0 \leq \prod_{k=1}^{n} (1 - {a \over k} ) \leq \exp (-a H_n) $$
where:
$$H_n = \sum_{k=1}^{n} \frac{1}{k}$$
Hence:
$$\lim_{n\to\infty} \prod_{k=1}^{n} (1 - {a \over k} ) = 0 $$
as $H_n \to \infty.$
