Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 3 / 12F / b / ii / Solution

False. Take the uniformly continuous function $f(x)=x$. Thenwhich is unbounded as a function of $x$ for every fixed $n$. Thus $g_n$ does not converge uniformly to $f^2$ on $\mathbb{R}$.