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I need help to prove this.

Let $f:\mathbb{R}\longrightarrow{\mathbb{R}}$ be an absolutely continuous function in any interval of the form $\left. [ -k,k \right ]$. If $f^{\prime}$ is in $L(\mathbb{R})$ and $\left\{{x_n}\right\}$ is a sequence that converges to infinity. Show that $\displaystyle\lim_{n \to{\infty}}{\displaystyle \left |{f(1+x_n)-f(x_n)}\right |}=0$

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closed as off-topic by Goos, Davide Giraudo, Shuchang, hardmath, AlexR Dec 13 '13 at 14:46

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If $f$ is absolutely continuous $[a,b]$, then f has a derivative $f′$ almost everywhere, the derivative is Lebesgue integrable, and $f(x) = f(a) + \int_a^x f'(t) \, dt$. –  Mhenni Benghorbal Dec 13 '13 at 9:30
    
What is the role in your hint of the form $\left. [ -k,k \right ]$ of the intervals? –  user95747 Dec 13 '13 at 10:33
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