Extension of Proof of Runge's theorem

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  • How does the condition on $b$ in equation $11.1$ guarantees the existence of number $r$,$0<r<1$ such that $\vert b-a \vert \ <r\vert z-a\vert \forall z\in K$?

Take $r\in\left(\frac{|b-a|}{d(a,K)},1\right)$. Then $r<1$ and, if $z\in K$,$$|b-a|<r.d(a,K)\leqslant r.d(a,z)=r.|z-a|.$$

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