Theorem (Nakai- Moishezon): A Cartier divisor $D$ on proper scheme $X$ is ample if and only if, for every integral subscheme $Y$ of $X$, one has $D^{\text{dim}(Y)}. Y > 0$
Defintion: Let $X$ be a complete scheme. A divisor $D$ on $X$ is said to be very ample if there exsts a closed immersion $X \hookrightarrow \mathbb{P}_{k}^{r} = \mathbb{P}$ such that $\mathcal{O}_{X}(D) \simeq \mathcal{O}_{\mathbb{P}}(1)|_{X}$. We apply the same terminology to the associated line bundle $L = \mathcal{O}_{X}(D)$.
Let $X$ be a smooth projective scheme and $Z$ a smooth projective subscheme of $X$ and $\pi: \widetilde{X} \longrightarrow X$ be the blow up of $X$ along of $Z$ with exeptional divisor $E$.
Let $H$ be an ample divisor on $X$.
How to use Nakai-Moishezon's theorem to show that the divisor $\pi^{*}mH - E$ is very ample on $\widetilde{X}$ for all integers $m$ sufficiently large ?
I would greatly appreciate any help.
Thank you very much.