Let $X$ be a projective variety, and $Z$ be a closed subscheme of $X$ of dimension $0$. Let $A$ be an ample Cartier divisor of $X$ and $\mathcal{I}$ be the ideal sheaf of $Z$. From the definition of ideal sheaf, we have an exact sequence of sheaves
$\begin{equation} 0 \to \mathcal{I} \to \mathcal{O}_X \to i_*\mathcal{O}_Z \to 0, \end{equation}$
where $i$ is the inclusion map $i: Z \to X$.
Tensoring with $\mathcal{O}_X(nA)$ for $n$ sufficient large, we obtain
$\begin{equation} 0 \to \mathcal{I}\otimes \mathcal{O}_X(nA) \to \mathcal{O}_X(nA) \to i_*\mathcal{O}_Z\otimes \mathcal{O}_X(nA)=i_*\mathcal{O}_Z \to 0. \end{equation} $
I wonder why the equality $i_*\mathcal{O}_Z\otimes \mathcal{O}_X(nA)=i_*\mathcal{O}_Z$ is true. I understand when $n$ is large enough, the very ampleness implies generated by global sections, but not sure what is the next step.
Thanks!