Let $\pi \colon X \to \mathbb{P}^n$ be a closed embedding given via an invertible sheaf $\cal L$ with global sections $s_0, \dots, s_n$. Thus ${\cal L} \cong \pi^* {\cal O}_X$.
Why is ${\cal L} \cong {\cal O}_X(1)$?
In other words, why is saying that an invertible sheaf gives a closed embedding the same as saying that it is isomorphic to ${\cal O}_X(1)$ (= being very ample)?
(This is a part of Exercise 16.6.A of Vakil's notes on algebraic geometry.)