$\forall x>0:f(nx)\mathrm{ converges}$, so does $f$ at infinity?

Asked Oct 14 '22 at 20:34

Active Oct 14 '22 at 20:34

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Suppose $f:\mathbb{R}_{>0}\to \mathbb{R}$ is continuous and $\forall x>0$ the sequence $\{f(nx)\}_{n=1}^{\infty}$ converges. Does $f$ have limit at infinity?

I tried first to show that $f$ is at least bounded and it sufficies to show that $\left\{\sup\limits_{n\in\mathbb{N}}\{f(n\cdot 1/t)\}_{n=1}^{\infty}\right\}_{t=1}^{\infty}$ is bounded, but I got nothing from this... Can you give any hint?

asked Oct 14 '22 at 20:34

matheg

This is a very popular application of Baire Category Theorem and it has been answered long back. – geetha290krm Oct 14 '22 at 23:29
1

@geetha290krm Is this comment helpful? You should mark it as a duplicate. – Kritiker der Elche Oct 14 '22 at 23:31
See https://math.stackexchange.com/questions/63870/a-classical-problem-about-limit-of-continuous-function-at-infinity-and-its-conne for example. – geetha290krm Oct 14 '22 at 23:34
Does this answer your question? A classical problem about limit of continuous function at infinity and its connection with Baire Category Theorem – Paul Frost Oct 14 '22 at 23:41
Does this answer your question? If $,\lim_{n\to\infty}f(nx),$ exists, for all $x\in\mathbb R$, then so does $,\lim_{x\to\infty}f(x),$ – Anne Bauval Oct 15 '22 at 06:18

$\forall x>0:f(nx)\mathrm{ converges}$, so does $f$ at infinity?

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