= Solution
{c}
Write $f_n\to f$ and $g_n\to g$ with <uniform convergence>. The <triangle inequality> gives
$$
\sup_E|(f_n+g_n)-(f+g)|\leq\sup_E|f_n-f|+\sup_E|g_n-g|\longrightarrow0.
$$
Hence $f_n+g_n\to f+g$ with <uniform convergence>.
If every $f_n,g_n$ is a <bounded function>, part <5/a/iii> gives constants $M_f,M_g$ bounding the two families. Their <pointwise limits> satisfy $|f|\leq M_f$ and $|g|\leq M_g$. Consequently
$$
\begin{aligned}
\sup_E|f_ng_n-fg|
&\leq\sup_E\bigl(|f_n|\,|g_n-g|+|g|\,|f_n-f|\bigr)\\
&\leq M_f\sup_E|g_n-g|+M_g\sup_E|f_n-f|\longrightarrow0.
\end{aligned}
$$
Thus \b[both sums and products have <uniform convergence> under the stated hypotheses]. The product estimate is the <uniform convergence product rule for bounded functions>.
Back to article page
