 Solution

The hypothesis puts zero in the common unbounded resolvent component from part (iii). Hence $0\in {\rho }_B(x)$ for $B=\overline{\mathbb{C}[1,x]}$, and $(-x{)}^{-1}\in B$, or equivalently $x^{-1}\in B$. By the definition of this norm closure, for each $n$ choose a polynomial $p_n$ with $\Vert p_n(x)-x^{-1}\Vert <1/n$. ThereforeThis is an approximation in the algebra norm, not merely a scalar approximation on the Banach algebra spectrum.