Solution
= Solution
Take the <holomorphic function> $f(z)=1/z$ on a neighborhood of the annulus. Every <polynomial> has <integral> zero around the unit circle by the <Cauchy integral theorem>, while $\int_{|z|=1}z^{-1}\,dz=2\pi i$. <Uniform convergence> on the annulus would imply convergence of these <contour integrals>, since their difference is bounded by $2\pi$ times the uniform error. Thus \b[uniform <polynomial> approximation on the full annulus need not exist].