Not a uniformly convergent polynomial sequence

Prove that there does not exist a sequence $\{ p_n(z)\}_{n \in \mathbb{N}}$ of complex polynomials such that $p_n(z) \rightarrow \frac{1}{z}$ uniformly on $\mathcal{C}_R=\{ z \in \mathbb{C} \mid \left| z \right| = R\}$ .

Solution

If such sequence existed then the convergence on the compact set $\left| z \right| = r$ would be uniform.

However,

$\begin{align*} 2 \pi i &= \oint \limits_{\mathcal{C}_R} \frac{{\rm d}z}{z} \\ &= \oint \limits_{\mathcal{C}_R} \lim_{n \rightarrow +\infty} p_n (z) \, {\rm d}z \\ &= \lim_{n \rightarrow +\infty} \oint \limits_{\mathcal{C}_R} p_n (z) \, {\rm d}z \\ &= \lim_{n \rightarrow +\infty} 0 = 0 \end{align*}$

which is an obscurity.

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Not a uniformly convergent polynomial sequence

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