calculus - Show that a recursive sequence converges

Let $a_1 = 1, a_2 = 2$ and $a_n = {1\over 2}(a_{n-1}+a_{n-2})$
Show that the sequence converges.

At first I thought using the theorem which says that a bounded and monotone sequence converges, but the sequence (at least the first terms) is not monotone.

I suspect I should use Cauchy's criteria but don't know how to apply it here.

Be glad for help.

Answer

We can prove that $$|a_n-a_{n-1}|=\frac{1}{2^{n-2}}$$

You can check that it works for $n=2$. Assuming it holds for $n$, we have $$|a_{n+1}-a_n|=\left|\frac{a_n+a_{n-1}}{2}-a_n\right|={1\over 2}|a_{n-1}-a_n|={1\over 2}|a_n-a_{n-1}|={1\over 2}\frac{1}{2^{n-2}}=\frac{1}{2^{n+1-2}}$$

This should lead to a proof of convergence.