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## Homework Statement

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.

## Homework 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?