calculus - Prove that $x_{n+2} := frac{1}{2}(x_n + x_{n+1})$ converges, if $x_0 = 1$ and $x_1 = 2$?

This question is related to my other question, where I had just to find the limit (which is $\frac{5}{3}$) of the following defined sequence:

$$x_0 = 1 \\ \\
x_1 = 2 \\ \\
x_{n + 2} = \frac{1}{2} (x_n + x_{n + 1})$$

Now, I need to prove that $\frac{5}{3}$ is really the limit, so I started my saying:

For any $\epsilon > 0$, there's an $N > 0$, such that if $n > N$, then $|x_n - \frac{5}{3}| < \epsilon$.

This means that we want to show that for any $n > N$, the distance between our limit $\frac{5}{3}$ and $x_n$ is less than $\epsilon$. Or, again, I need to show: $$0 <|x_n - \frac{5}{3}| < \epsilon$$ for all $n$ greater than a certain $N$.

Now, how do I proceed proving this limit?