I am looking for some references for the following statement:

Let $G$ be a linearly reductive algebraic group acting on a quasi-projective scheme $X$, over an algebraically closed field $K$. Let $L$ be an ample line bundle on $X$. Then there exists a sufficiently large $l>>0$ such that $L^{\otimes l}$ is $G$-linearized.

This one was suggested to me by a friend of mine, but he doesn't know any references. I tried to search it on some classic books, as GIT or Newstead (Introduction to moduli and orbit spaces), but without any results. If you know another similar statement in a book or a reference where I can find also a general treatment, I would be very happy.

Thank you!

P.S.: I trust that this is true, but of course I am not so sure. I am looking for a reference also to see if the hypothesis are right or no.