Given a real square matrix $A$ with its spectral radius $\rho(A)<1$, it seems well-known [REF] that there exist $c \geq 1$ and $\rho_\star \in [\rho(A),1)$, such that $$ \|A^k \|_2 \leq c \rho_\star^k, $$ for any non-negative integer $k$, where $\|\cdot\|_2$ denotes the spectral norm.
I am wondering how can this be proved. Moreover, can we always choose $\rho_\star = \rho(A)$?
This statement (in the case that $\rho_\star > \rho(A)$ is a consequence of Gelfand's formula, which states that $$ \lim_{k \to \infty}\|A^k\|^{1/k} = \rho(A). $$ By the definition of a limit, this implies that for any $\rho_\star \in (\rho(A),1)$, the sequence $x_k = \|A^k\|^{1/k}$ will satisfy $x_k < \rho_\star$ whenever $k > K$ for some sufficiently large $K \in \Bbb N$. Thus, for all such $k$, we have $$ \|A^k\|^{1/k} \leq \rho_\star \implies \|A^k\|^{1/k} \leq \rho_\star^k. $$ The constant $c$ allows us to extend this result to $k = 1,\dots,K$. In particular, it suffices to take $$ c = \max\left\{\max_{k = 1,\dots,K} \frac{\|A^k\|^{1/k}}{\rho_\star^k}, 1\right\}. $$ Note that we are taking a maximum of finitely many values, so this maximum is well defined.