Suppose that $f$ is analytic in the annulus $1<|z|<2$ and there exist a sequence of polynomials converging to $f$ uniformly on every compact subset of this annulus. Show $f$ has an analytic extension to all of the disc $|z|<2$.
Take the Laurent series for $f$ in the annulus and calculate the coefficients of negative powers using the polynomial approximation.
Conclude that the Laurent series is a power series.
A power series that converges in the annulus already converges in the disc.