recursive sequence convergence

antiemptyv

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

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

2. Relevant equations

Since this problem comes from the section on Cauchy sequences, I assume we will need to show [tex](x_n)[/tex] 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.

[tex]|x_{n+1}-x_n| = |\frac{1}{3}x_n + 2 x_{n-1} - x_n | = \frac{2}{3}|x_{n-1} - x_n|[/tex]

Thus [tex](x_n)[/tex] is contractive, so it is convergent.

antiemptyv

OK. so now how do we go about finding the limit of [tex](x_n)[/tex]?