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?
How to Prove that a sequence $\{f_n\}$ of functions converges uniformly?
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calculus
real-analysis
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0Ok , thanks alot – 2012-04-01
1 Answers
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In fact, we never have the uniform convergence on $(0,+\infty)$, since if $f(x_0)\neq 0$ then $\sup_{x>0}|f(nx)|\geq \left|f\left(n\frac{x_0}n\right)\right|=|f(x_0)|>0$.