The relationship between $\|A^n\|,\|e^{tA}\|$ and its spectral radius and spectral abscissa

Question

Define the spectral radius of a matrix $A\in\mathbb{C}^{m\times m}$ $\rho(A)=\max\limits_i|\lambda_i|$, where $\lambda_i$ is $A$'s eigenvalue; the spectral abscissa $\alpha(A)=\max\limits_i \Re(\lambda_i)$.

Prove

1. $\lim\limits_{n\rightarrow+\infty}\|A^n\|=0\iff \rho(A)<1$
2. $\lim\limits_{t\rightarrow+\infty}\|e^{tA}\|=0\iff \alpha(A)<0$