If $\{f_n\}$ converges uniformly to $f$ and $\{g_n\}$ converges uniformly to $g$, does it mean $\{f_ng_n\} $ will converge uniformly to $fg$?
I am absolutely stuck on this. Please help.
2026-04-03 05:20:31.1775193631
If $\{f_n\}$ and $\{g_n\}$ be uniformly convergent sequences of bounded functions on S, then $\{f_ng_n\}$ is uniformly convergent on S.
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Hint
\begin{align}||f_ng_n-fg||_\infty&\le||f_ng_n-f_ng+f_ng-fg||_\infty\\&\le||f_n||_\infty||g_n-g||_\infty+||g||_\infty||f_n-f||_\infty\end{align} and use the hypothesis that those functions are bounded.