Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics II / 5 / b / ii / Solution

On $E=\mathbb{R}$, take $f_n(x)=x$ and $g_n(x)=1/n$. Then $f_n\to f(x)=x$ with uniform convergence, because there is no error, and $g_n\to 0$ with uniform convergence, with error $1/n$. Their products $f_n{g}_n=x/n$ converge pointwise to zero, but for every $n$ the choice $x=n$ gives $|f_n(x)g_n(x)|=1$. Thus the products do not converge uniformly. Here the first family consists of unbounded functions, exactly the missing hypothesis in the preceding estimate.