recursive sequence convergence

[antiemptyv]

1. The problem statement, all variables and given/known data

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.

2. Relevant 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)?