= Solution
The complement of a <compact> planar set has a unique unbounded component: outside a large disc lies a connected exterior, and every unbounded component meets that exterior. The <Neumann series> for $|\lambda|>\|x\|$ converges in both algebras, so this exterior belongs to $\rho_B(x)$. Part i retains the whole component of $\rho_A(x)$ containing it. That retained component is unbounded and therefore is also the unique unbounded component of $\rho_B(x)$. \b[The two unbounded components coincide.]
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