Let \(\mathcal{F}\) be a bounded family of \(n\times n\) complex matrices and \(\Sigma_k(\mathcal{F})=\{A_1\cdots A_k\;:\;A_i\in \mathcal{F}\}\). For each \(k\geq 1\), define the number \(\bar{\rho}_k(\mathcal{F})=\sup_{Q\in \Sigma_k(\mathcal{F})} \rho(Q)\) where \(\rho(\cdot)\) denotes the spectral radius of a matrix. The generalized spectral radius of \(\mathcal{F}\) is \(\rho(\mathcal{F})=\limsup_{k\rightarrow\infty} \bar{\rho}_k(\mathcal{F})^{1/k}\). \(\mathcal{F}\) is said to be asymptotically regular if \[ \rho(\mathcal{F})=\lim_{k\rightarrow\infty} \bar{\rho}_k(\mathcal{F})^{1/k}. \] \(\mathcal{F}\) is said to have finiteness property if there exists \(k^*\geq 1\) and a \(P\in \Sigma_{k^*}(\mathcal{F})\) such that \[ \rho(\mathcal{F})=\bar{\rho}_{k^*}(\mathcal{F})^{1/{k^*}}=\rho(P)^{1/k^*} \] and the special \(P\) is called a spectrum-maximizing product for \(\mathcal{F}\). Theorem 3.1 states a sufficient condition on a matrix \(P\) and \(k^*>1\) such that \(\lim_{k\rightarrow\infty} \bar{\rho}_k(\mathcal{F})^{1/k}\geq \rho(P)^{1/k^*}\). As a consequence, Corollary 3.1 states a sufficient condition on a \(\mathcal{F}\) with finiteness property to be asymptotically regular. At the end of the article, there is an application Theorem 5.2 which states that if \(\mathcal{F}\) is a bounded family of nonnegative matrices with finiteness property and there exists a primitive spectrum-maximizing product, then \(\mathcal{F}\) is asymptotically regular.