Bound on the norm of a matrix power

Question

Suppose we have the square matrix $A$ and we know that its spectral radius $\rho(A)$ is less than $1$, therefore matrix $A$ is stable. How can we prove that $\exists \gamma \in(0,1)$ and $\exists M >0$ such that $$\|A^k\|\leq M\gamma^k, \:\:\:\: \forall k\geq0$$ What I tried so far is $\|A^k\|=\|A\dots A\|\leq\|A\|\dots \|A\| =\|A\|^k$ so taking $\gamma=\|A\|$ I should be close to the above inequality, but I am not sure it is correct.