# Real analysis: Sequences question

1. Apr 13, 2009

### jinbaw

1. The problem statement, all variables and given/known data
If Xn is bounded by 2, and $$|X_{n+2} - X_{n+1}| \leq \frac{|X^2_{n+1} - X^2_n|}{8}$$, prove that Xn is a convergent sequence.

2. Relevant equations

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

Last edited: Apr 13, 2009
2. Apr 13, 2009

### Billy Bob

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

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

3. Apr 13, 2009

### 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.

4. Apr 13, 2009

### Dick

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

5. Apr 13, 2009

### Billy Bob

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