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Let $f(x)$ be a continuous and strictly increasing function, for all $x\in \mathbb R$, and $f'(x)\leq C$, for all $x\in \mathbb R$, and $C>0$. If $\{a_{n}\}_{n\in \mathbb Z}$ is an increasing sequence of real numbers with $a_{n}\to \pm\infty$ as $n\to \pm\infty$. Is the following true: Fix $a\in \mathbb R$, then we can find $r$ very large (independent of $a$) so that

$$\sum_{a_{n}\notin [a-r,a+r]}f'(a_{n})< \infty$$

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LOL, I guess no! If we take $f(x)=x$. –  Emily May 7 '12 at 21:40
    
May be it is worth to delete this question? –  no identity May 7 '12 at 21:42
    
Yes, I was thinking in something else. –  Emily May 7 '12 at 21:44

1 Answer 1

Hint:

Consider $f(x)=x$, $a_n=n$

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