  Let $f$ be a holomorphic function on the unit disc, and $(S_{n_{k}})$ be a
subsequence of its Taylor polynomials about $0$. It is shown that the
nontangential limit of $f$ and lim$_{k\rightarrow \infty }S_{n_{k}}$ agree at
almost all points of the unit circle where they simultaneously exist. This
result yields new information about the boundary behaviour of universal Taylor
series. The key to its proof lies in a convergence theorem for harmonic
measures that is of independent interest.
