How to prove the limit of the sequence?

[kakarotyjn]

Homework Statement

Assume the sequence $$\{ a_n \}$$ satisfys $$0 < a_1 < 1$$ and $$a_{n + 1} = a_n (1 - a_n )(n \ge 1)$$,prove $$\mathop {\lim }\limits_{n \to \infty } {\kern 1pt} {\kern 1pt} {\kern 1pt} na_n = 1$$

Homework Equations

The Attempt at a Solution

a_n is a monotonicly decreasing sequence,and it is above zero,so a_n has a limit,zero.

let $$b_n=n*a_n$$ then we got $$\frac{{b_{n + 1} }}{{n + 1}} = \frac{{b_n }}{n}(1 - \frac{{b_n }}{n})$$,but I don't know how to do next.

 

[hunt_mat]

I would go about showing first that a_n is always positive and decreasing. I would then ewxamine if you b_n is an increasing or decreasing sequence. After thay I might examine the following:

$$|b_{n}-1)|$$

to see what I could find.