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If a sequence of functions $f_j$ on a domain $S \subseteq \mathbb{R}$ has the property that $f_j \rightarrow f$ uniformly on $S$ then does it follow that $(f_j)^2 \rightarrow f^2$ uniformly on $S$?

I know this to be false. Suppose $f_j(x) = x + (1/j)$ for example.

But what would make this statement true? What if $f$ is bounded? Does anyone have a proof to show that if $f$ is bounded then the above is true?

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    Well it would certainly be true for $S$ compact, as on a compact set convergence $\implies$ uniform convergence. The same proof should work for bounded.2012-02-29

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