SCHWEITZER 2000 Problem 7

Let $H(D)$ denote the space of functions holomorphic on the disc $D=\{ z\colon |z|<1 \}$ , endowed with the topology of uniform convergence on each compact subset of $D$ . If $f(z)=\sum_{n=0}^{\infty} a_nz^n$ , then we shall denote $S_n(f,z)=\sum_{k=0}^n a_kz^k$ . A function $f\in H(D)$ is called universal if, for every continuous function $g\colon\partial D\rightarrow \mathbb{C}$ and for every $\varepsilon >0$ , there are partial sums $S_{n(j)}(f,z)$ approximating $g$ uniformly on the arc $\{ e^{it} \colon 0\le t\le 2\pi - \varepsilon\}$ . Prove that the set of universal functions contains a dense $G_{\delta}$ subset of $H(D)$ .