If $f(nc) \to 0$ for all $c>0$ then prove that $f(x) \to 0$ as $x \to \infty$

Asked Oct 24 '22 at 21:21

Active Oct 24 '22 at 21:21

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Suppose $f$ is a continuous function on $\Bbb R$. If $f(nc) \to 0$ as $n \to \infty$ for all $c>0$ then prove that $f(x) \to 0$ as $x \to \infty$

I have solved the problem after extending the function continuously to $\hat{ \Bbb R}=\Bbb R \cup \infty$ which is complete then $nc$ is a subsequence going to $\infty$ and hence $f(x) \to 0$.

I think there might be some problem with this solution. Is there some elementary way(highly relative, so you can give any solution that comes up to your mind first)?

asked Oct 24 '22 at 21:21

Ri-Li

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How do you know you can extend this function continuously to $\hat{\Bbb{R}}$? – Theo Bendit Oct 24 '22 at 21:25
I feel somehow I have to use Baire Category. Is there any other approach more elementary? – Ri-Li Oct 24 '22 at 22:21
@Ri-Li: One of the proof has a reference to a set of problems by Makarov (there is an English translation by the AMS) that uses just simple properties of the real line. – Mittens Oct 25 '22 at 00:35
Can you send me the link? – Ri-Li Oct 25 '22 at 18:35

If $f(nc) \to 0$ for all $c>0$ then prove that $f(x) \to 0$ as $x \to \infty$

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