= Uniform convergence product rule for bounded functions
{title2=$\|f_ng_n-fg\|_\infty\leq M_f\|g_n-g\|_\infty+\|g\|_\infty\|f_n-f\|_\infty$}
If <sequences of functions> $f_n\to f$ and $g_n\to g$ have <uniform convergence> on the same <domain> and each term is a <bounded function>, their products converge uniformly to $fg$. The two families are <uniformly bounded families of functions>: one late term and the <Uniform Cauchy criterion> bound the entire tail, while finitely many early terms have a finite common bound. Write $f_ng_n-fg=f_n(g_n-g)+g(f_n-f)$ and take the <supremum norm> to obtain the displayed estimate. The boundedness hypothesis cannot be dropped: $f_n(x)=x$ and $g_n(x)=1/n$ converge uniformly on $\mathbb R$, whereas $x/n$ fails to converge uniformly to zero.
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