An equivalent statement for convergence

Asked Jun 21 '12 at 11:11

Active Jun 21 '12 at 12:15

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Possible Duplicate:
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?

edited Jun 12 '20 at 10:38

Community

asked Jun 21 '12 at 11:11

FUZxxl

9,307

it has been posted before: http://math.stackexchange.com/questions/63870/a-classic-problem-about-limit-of-continuous-function-at-infinity-and-its-connect – clark Jun 21 '12 at 11:15
@clark Didn't knew that. Thanks for pointing that out! Please close my question. – FUZxxl Jun 21 '12 at 11:32
no problem! I am quite new here I do not know how to do that... – clark Jun 21 '12 at 11:34
I couldn't have guessed that the answer is positive! – Siminore Jun 21 '12 at 12:17

An equivalent statement for convergence

Prove or disprove

0 Answers0