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Give an example of a sequence of functions $\{f_n(x)\}$ that converges uniformly on the set of the real numbers but $(f_n(x))^2$ does not converge uniformly uniformly on the set of real.

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To make this a better question, please let us know (1) how you encountered the problem and (2) what you have tried so far. This will help others write helpful answers. Also, if you are using a particular textbook, it helps to indicate which one. – Carl Mummert Sep 30 '11 at 2:03
Moreover, please do not post in the imperative (e.g. like you are assigning us all homework). – Austin Mohr Sep 30 '11 at 2:32
Is that example corresponds to your question? $1+arctg(x)^n$ – Artur Mustafin Sep 30 '11 at 2:39
@Artur: that sequence does not converge uniformly. Forgive me if that is your point; whatever your comment referred to must have been deleted. – robjohn Sep 30 '11 at 9:36

Try $f_n(x)=1/x+1/n$ for $x\not=0$ and $f_n(0)=1/n$.

You could even try $f_n(x)=x+1/n$.

What is it that seems to make these work? That is, what characteristic of these functions keeps $f_n^2(x)$ from converging uniformly?

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For $f_n^2$, the rate of convergence is 2 when $x=0$, whereas it is 1 everywhere else, thus convergence is not uniform.

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