Recursive sequence convergence

[antiemptyv]

Homework Statement

Let $$x_1 < x_2$$ be arbitrary real numbers and let $$x_n :=\frac{1}{3}x_{n-1} + \frac{2}{3}x_{n-2}$$. Prove the sequence $$(x_n)$$ converges.

Homework Equations

Since this problem comes from the section on Cauchy sequences, I assume we will need to show $$(x_n)$$ is a Cauchy sequence. I'm not so well-versed in working with the recursive sequences especially with arbitrary initial values.

Any advice on getting started?

 

[antiemptyv]

would this be a valid solution? it looks like i can show the sequence is contractive.

$$|x_{n+1}-x_n| = |\frac{1}{3}x_n + 2 x_{n-1} - x_n | = \frac{2}{3}|x_{n-1} - x_n|$$

Thus $$(x_n)$$ is contractive, so it is convergent.

 

[antiemptyv]

OK. so now how do we go about finding the limit of $$(x_n)$$?