Show the sequence $a_n$ with $a_1 = 10$ and $a_{n+1} = \frac{1}{2}a_n+2$ is convergent

Question

Show the sequence $a_n$ with $a_1 = 10$ and $a_{n+1} = \frac{1}{2}a_n+2$ is convergent

I'm not sure how to solve this. I've tried a couple different things.

1. I believe the sequence is bounded below by 2 as $a_n \ge 0$ (not sure how to prove this)? If I could should the sequence is decreasing then I would have convergence (again not sure how to prove this. I tried $\frac{a_{n+1}}{a_n}$ but did not progress anywhere).

2. Maybe I could prove this sequence is Cauchy and thus converges? (not sure how to do this either).

Thanks.

Answer 1

$a_{n+1}-4=(a_n-4)/2$, so $a_n-4=(a_1-4)/2^{n-1}$ follows by induction.