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A classic problem about limit of continuous function at infinity and its connection with Baire Category Theorem

Someone gave me this little question:

Prove or disprove

For a continuous function $f:\mathbb R\to\mathbb R$ the statement $\lim_{x\to\infty}f(x)=0$ holds, iff for each $\varepsilon>0$ the statement $\lim_{n\to\infty}a_n=0$ holds, where $a_n=f(n\cdot\varepsilon)$.

I thought about it for some time, but I failed to find a proof or a counterexample. So, how can I (dis)prove this statement?

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    it has been posted before: http://math.stackexchange.com/questions/63870/a-classic-problem-about-limit-of-continuous-function-at-infinity-and-its-connect2012-06-21
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    @clark Didn't knew that. Thanks for pointing that out! Please close my question.2012-06-21
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    no problem! I am quite new here I do not know how to do that...2012-06-21
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    I couldn't have guessed that the answer is positive!2012-06-21

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