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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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    @Artur: that sequence does not converge uniformly. Forgive me if that is your point; whatever your comment referred to must have been deleted.2011-09-30

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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.