How to prove the limit of the sequence?

[kakarotyjn]

Homework Statement

Assume the sequence \{ a_n \} satisfys 0 &lt; a_1 &lt; 1 and <br /> a_{n + 1} = a_n (1 - a_n )(n \ge 1),prove <br /> \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 <br /> \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:

<br /> |b_{n}-1)|<br />

to see what I could find.