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> Show the sequence $a\_n$ with $a\_1 = 10$ and $a\_{n+1} = \frac{1}{2}a\_n+2$ is convergent
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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.