A bounded sequence satisfying $2a_n \le a_{n+1} +a_{n-1}$ for all n converges.

Question: Let $(a_n)$ be a bounded real sequence such that $2a_n \le a_{n+1} +a_{n-1}$ for all positive integers n. Let $b_n = a_{n+1} - a_n$.

Show that $(b_n)$ converges to 0 and that $(a_n)$ converges.

I have shown that $(b_n)$ is a monotone increasing sequence that is bounded above and so it must converge. However, I am struggling to show that $(b_n)$ converges to 0. I have tried using the fact that $b_{n} +b_{n-1}$ = $a_{n+1} - a_{n-1}$ but I really don't think this is getting me anywhere. I would appreciate some hints as to how to actually show $(b_n)$ converges to 0 as well as how to prove that, as a consequence, $(a_n)$ converges.

Solution 1:

Suppose $b_n \to b$. If $b >0$ then $a_{n+1}-a_n \to b$ and $a_{n+m}-a_m=b_{n+m}+b_{n+m-1}+...b_{m+1}>(b-\epsilon)+(b-\epsilon)+...+(b-\epsilon)$ ($m$ terms) if $m$ is sufficienlty large. This makes $(a_n)$ unbounded, so $b$ cannot be $>0$. A similar argument shows $b$ cannot be $<0$ so $b=0$.

Now $b_n$ is increasing to $0$ so $b_n \leq 0$ for all $n$. This makes $a_n$ decreasing. It is also bounded, hence convergent.