Norm of a matrix exponential

Question

Can any one prove the following inequality $$||e^{Pt}||\leq e^{t\alpha{(P)}}\sum_{k=0}^{r-1}\frac{(||P||\sqrt{r}\,t)^k}{k!}$$, where $r$ is the order of the matrix $P$ and $\alpha(P)$ be the maximum eigenvalue of the matrix $P.$ With details.