I came across with a statement
Let $X\rightarrow \mathbb{P}^n$ be a map defined by a linear system $|L|$ for some line bundle $L$ on $X$. It is embeding if $H^0(\mathbb{P}^n,O(1))\rightarrow H^0(X,O(1))=H^0(X,L)$ is injective.
Here $O(1)$ stands for the line bundle defined by hyperplane on $\mathbb{P}^n$, and the induced one on $X$ (hence it coincides with $L$).
The statement above means that if global sections of $O(1)$ can be distinguished after restricting to the image of $X$, then $X\rightarrow \mathbb{P}^n$ is embedding. It seems true but I am not so convinced. How should I understand the statement above?