Real analysis: Sequences question

[jinbaw]

Homework Statement

If X_n is bounded by 2, and $$|X_{n+2} - X_{n+1}| \leq \frac{|X^2_{n+1} - X^2_n|}{8}$$, prove that X_n is a convergent sequence.

Homework Equations

The Attempt at a Solution

I believe the solution lies in proving X_n a Cauchy sequence, but I'm not sure how to work it out. I considered |X_n - X_m| adding and subtracting the terms between m and n but i got stuck.
I also tried to check for telescoping, with no luck.

 

[Billy Bob]

Factor $$X^2_{n+1} - X^2_n$$.

Then get an upper bound for $$\frac{|X^2_{n+1} - X^2_n|}{8}$$.

 

[jinbaw]

$$\frac{|X^2_{n+1} - X^2_n|}{8} = \frac{|(X_{n+1} + X_n)(X_{n+1} - X_n)|}{8} \leq \frac{|X_{n+1} - X_n|}{2}$$

Iterating, I reached $$|X_{n+2} - X_ {n+1}| \leq \frac{|X_2 - X_1|}{2^n}$$
I'm not sure if I continued in the right track.. but I'm stuck.
Thanks for your input.

 

[Dick]

Now go back to trying to show the sequence is Cauchy by adding and subtracting terms in |xn-xm|. Use the triangle inequality.

 

[Billy Bob]

Or could even look at the sequence of partial sums, getting it bounded above.