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Suppose that the sequences $(f_n)$ and $(g_n)$ converge uniformly to respective functions $f$ and $g$ on some domain $S$. Can someone provide me with a counterexample (preferably with proof) of sequences $(f_n)$ and $(g_n)$ converge uniformly to $f$ and $g$ on $S$ but $(f_ng_n)$ does NOT converge uniformly to $fg$ on $S$.

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Consider $f_n,g_n:\mathbb{R}\rightarrow \mathbb{R}$ that send $x$ to $x+\frac{1}{n}$. Clearly $f_n,g_n$ converge uniformly to $f(x)=x$, but $f_ng_n$ does not converge uniformly to $x^2$.

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