Let $\{x_n\}_{n=-\infty}^{\infty}$ be a positive sequence decreasing to zero as $|n| \to \infty$.
Show there is a sequence $\{y_n\}$ satisfying \begin{align} y_n >& x_n \tag{1}\\ y_{n-1}+y_{n+1}-2y_n \ge& 0 \tag{2} \end{align}
I have attempted this question with this approach where $y_n=c_n$.
The method gives me (1).
But not being able to get $\frac{1}{n} \ge c_n-c_{n+1} \ge \frac{1}{n+1}$, the method fails for (2).