How to prove spectral radius of $A-A^{\infty}$ is less than 1? $A$ is semi convergent

Question

Suppose that $A$ is a semi-convergent matrix. Denote the limiting matrix as $A^{\infty}$. Can someone tell me, why the spectral radius of ($A-A^{\infty}$) is less than 1? I guess it is because $A-A^{\infty}$ is a convergent matrix (hence its spectral radius is less than 1). But how to prove $A-A^{\infty}$ is a convergent matrix? OR is $\rho(A-A^{\infty})<1$ a part of the implication of the-semi-convergence of $A$?