Note: This is a homework problem.

Suppose that {$\displaystyle {a_n}$} is a bounded sequence of reals such that for any n, $\displaystyle a_n \le \frac{a_{n-1} + a_{n+1}}{2} $.

Show that $\displaystyle b_n = a_{n+1} - a_n$ is an increasing sequence and, $\displaystyle {a_n}$ converges.

I've already done the first part, that $\displaystyle {b_n}$ is an increasing sequence. For the second part, my attempt so far has been that looking at $\displaystyle a_2 \le \frac{a_1 + a_3}{2} $, there are two possibilities:

Case 1: It's possible that $\displaystyle a_3 \ge a_2 $. In that case, if for some integer k, $\displaystyle a_{k+1} \ge a_k $, then we can use that $\displaystyle a_{k+1} \le \frac{a_k + a_{k+2}}{2}$ to show that $\displaystyle 2*a_{k+1} \le a_k + a_{k+2} \le a_{k+1} + a_{k+2} $, and so $\displaystyle a_{k+1} \le a_{k+2} $, establishing by induction that {$\displaystyle a_n$} is also an increasing bounded sequence and then convergent.

Case 2: It's also possible that $\displaystyle a_1 \ge a_2 $. This is where I'm having trouble. The only way I can see to continue with this is to show that the sequence {$\displaystyle a_n$} is decreasing in this case. After several failed attempts, I'm starting to become skeptical that it actually is decreasing. Is this not going to work, and should I abandon this attempt? If so, does anyone have a suggestion for what direction to go in?