I come across with this question:
Consider $\{a_n\}$ is a bounded sequence, $\lim_{n\rightarrow \infty} (a_n - 2a_{n+1}+a_{n+2}) = 0$, prove that $$ \lim_{n\rightarrow \infty}(a_n - a_{n+1}) = 0 $$
I can only prove that $b_n = a_n - a_{n+1}$ is Cauchy, how can I conclude that the limit is $0$?
Suppose $a_n -a_{n+1} \to l>0$. Then $a_n -a_{n+1} >\frac l 2$ for $n$ suffcienlty large, say for $n \geq n_0$. So $a_{n+1} <a_n-\frac l 2$ for $n \geq n_0$. Iterate this to get $a_{n+1} <a_{n_0} -\frac {(n+1-n_0) l} 2$ and conclude that $a_n \to -\infty$ as $n \to \infty$ contradicting the fact that $(a_n)$ is bounded.
Similarly, $a_n -a_{n+1} \to l<0$ leads to a contradiction. Hence, $a_n -a_{n+1} \to 0$.
