= Moving an exterior pole to obtain pointwise polynomial approximation
{title2=$p_n(z)\to(z-z_0)^{-1}$}
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.
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