Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 2H / i / Solution

True when “uniform” means on all of the unbounded quadrant $\Omega$. A uniformly convergent sequence $(P_n)$ is uniformly Cauchy. For sufficiently large $m,n$, the polynomial $P_n-P_m$ is bounded on $\Omega$. Every nonconstant polynomial is unbounded along some ray contained in $\Omega$, so $P_n-P_m$ is constant. Fixing one large $m$, the limit is therefore $P_m$ plus the limit of constants, hence is a polynomial.