Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 2F / ii

True. By the Weierstrass approximation theorem, choose a polynomial $Q_n$ with ${\sup }_{|t|\le n}|Q_n(t)-f(t)|<1/n$. For each fixed $t$, this bound applies once $n\ge |t|$, so $Q_n(t)\to f(t)$. This proves pointwise convergence; the same construction even gives uniform convergence on every closed interval.