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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).

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    @RobertIsrael I see your point, I relax this requirement in the question. However, as $x_n$ is a decreasing sequence, I think your example doesn't apply, but your point is still valid.2012-10-16

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$y_n=x_0+2^n\qquad (n\in\mathbb Z)$

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    I amended the title to match the question. Thank you very much.2012-10-16