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Show that $$f_n: \mathbb{R} \rightarrow \mathbb{R}; \ n\ge1; \ f_n (x)=\frac{x \sqrt{n}}{n \sqrt{n}+x^2}$$ does not converge uniformly on $\mathbb{R}$

I have showed that $f_n(x)$ pointwise converges to $0$.

Then I find the maximum of $|f_n(x)|$ and it is for $x=n^{3/4}$ and $x=-n^{3/4}$

Hence $\sup_{\mathbb{R}}{|f_n(x)|}=f_n(n^{3/4})=\dfrac{1}{2 n^{1/4}} \rightarrow 0$

Where is the mistake?

  • 0
    then is wrong the text of the exercise....2012-12-06
  • 0
    A [related problem](http://math.stackexchange.com/questions/370023/how-to-prove-a-sequence-of-a-function-converges-uniformly/370071#370071).2013-06-10

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