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If $f\colon (0,+\infty)\to\mathbb R$ is not identically $0$ and $$\lim _{x \to +\infty} f(x) = 0, $$ then does the sequence of functions $\{f_n\}$ defined by $$f_n(x) = f(nx)$$ converge uniformly to the zero function?

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On which set do you ask uniform convergence? If it's $(0,+\infty)$ it's not true, taking $f(x)=1/x$. – Davide Giraudo Apr 1 '12 at 13:14
Yes from (0, oo) but can you explain me Why its not true ? – عبدالرازق حاج يحيى Apr 1 '12 at 13:17
It satisfies the first condition, but $\sup_{x\in (0,+\infty}|\frac 1{nx}-0|$ is not finite. – Davide Giraudo Apr 1 '12 at 13:20
Ok , thanks alot – عبدالرازق حاج يحيى Apr 1 '12 at 13:21

1 Answer 1

up vote 4 down vote accepted

In fact, we never have the uniform convergence on $(0,+\infty)$, since if $f(x_0)\neq 0$ then $\sup_{x>0}|f(nx)|\geq \left|f\left(n\frac{x_0}n\right)\right|=|f(x_0)|>0$.

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