Sequences / Real Analyses question

[jinbaw]

Sequences / Real Analysis question

Homework Statement

a,b are the roots of the quadratic equation x² - x + k = 0, where 0 < k < 1/4.
(Suppose a is the smaller root). Let h belong to (a,b). The sequence x_n is defined by:
$$x_1 = h, x_{n+1} = x^2_n + k.$$

Prove that a < x_n+1 < x_n < b, and then determine the limit of x_n.

Homework Equations

The Attempt at a Solution

I have no idea how to start, if you could help me.
Thanks.

 

[mjsd]

perhaps starts by determining a and b in terms of k?

 

[jinbaw]

Okay, so I got $$a = \frac{1 - \sqrt{1 - 4k}}{2}, b = \frac{1 + \sqrt{1 - 4k}}{2}$$.

And I was able to prove $$X_{n+1} < X_n$$ by induction. But, I'm stuck on the outer inequalities.

EDIT: $$X_{n+1} < X_n$$ means that X₁ = h is the largest value of X_n for all n. And h belongs to (a,b), so X₁ < b, and consequently X_n < b.

I still need to prove that a is a lower bound..

 

[jinbaw]

I think the basic idea is:
$$X_{n+1} < X_n \Leftrightarrow X^2_n - X_n + k < 0$$

Therefore, X_n must be between the roots for this equation to be negative.
But is there a more mathematical way to state it?

 

[Billy Bob]

Good job! How about saying x^2-x+k=(x-a)(x-b) which is negative if and only if a<x<b.

 

[jinbaw]

Oh right! Thanks a lot :)