Moving an exterior pole to obtain pointwise polynomial approximation ($p_n(z)\to (z-z_0{)}^{-1}$)

Let $z_0$ be a boundary point of a compact convex set and choose exterior points ${\zeta }_n\to z_0$. Approximate $(z-{\zeta }_n{)}^{-1}$ uniformly on the compact set by a polynomial with error at most $1/n$. At each fixed $z\ne z_0$, the reciprocal converges and so does the approximating polynomial. On a boundary accumulating at $z_0$, uniform convergence to the singular reciprocal is impossible because each polynomial is bounded there. This distinguishes pointwise convergence from uniform convergence without confusing the two norms.