How do we show the limit? We have that $(a_n)$ is a bounded sequence of real numbers that satisfy $2a_{n+1}\leq a_n+a_{n+2}, \forall n\in \mathbb{N}$. 
I want to show that the sequence $b_n=a_{n+1}-a_n$ converges and that the limit is $0$. 
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I have shown that $b_n$ is increasing and bounded. 
So, $b_n$ converges. 
How do we find the limit? 

Note that $$a_{n+2}-a_{n+1}\ge a_{n+1}-a_n \,\,\,\,\forall n\in\mathbb{N}.$$ First
assume that $a_0\lt a_1,$ then the sequence $(a_n)$ is increasing and bounded. Therefore it is convergent.  
Next
if $a_0\gt a_1,$ Consider the sequence $(-a_n).$
In both cases $\lim b_n=0.$
