Uniform convergence product rule for bounded functions ($\Vert f_n{g}_n-fg{\Vert }_{\infty }\le M_f\Vert g_n-g{\Vert }_{\infty }+\Vert g{\Vert }_{\infty }\Vert f_n-f{\Vert }_{\infty }$)

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 $f_n{g}_n-fg=f_n(g_n-g)+g(f_n-f)$ 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.