Proof of a bounded below recursive sequence

Question

Say $a_1 = 8$ and $a_{n + 1} = \frac{1}{2}a_{n} + 2$. Prove $a_n$ is bounded from below. I use proof by induction to show that $a_n$ is monotone decreasing. Take n = 1: since $a_1 = 8$ > $a_2 = 6$. Thus, n = 1 is true. Then I suppose $a_k < a_{k+1}$ for all $k > 1$, I got $a_{k + 2} < a_{k + 1}$. That's all I got for this proof. If I need to prove a sequence is bounded below, I need to find a limit. How can I find the limit?

Answer 1

Prove, by induction, that $a_n \geq 0$ for all $n$. In fact $a_n \geq 2$ for all $n$.