Properties of dominant morphism of schemes I am trying to solve the Exercise 4.11, p.67 from Qing Liu's book Algebraic Geometry and Arithmetic Curves.
Let $f:X\to Y$ be a morphism of irreducible schemes with respective
generic points $\xi_X$, $\xi_Y$. We say that $f$ is dominant if $f(X)$ is dense in $Y$. Let us suppose that $X$, $Y$ are integral. Show that the following properties
are equivalent:
(i) $f$ is dominant;
(ii) $f^\#:\mathcal{O}_Y\to f_*\mathcal{O}_X$ is injective;
(iii) for every open subset $V$ of $Y$ and every open subset $U\subset f^{-1}(V)$, the map $\mathcal{O}_Y(V)→\mathcal{O}_X(U)$ is injective;
(iv) $f(\xi_X)=\xi_Y$;
(v) $\xi_Y ∈f(X)$.
I know how to prove the equivalence of (i), (iv) and (v) but I had some troubles while was trying to prove the equivalence of (i) and (ii). What is the point here?

To prove $i) \Rightarrow ii)$ use the fact that $f(\xi_X)=\xi_Y$. 
You can suppose $f^{\sharp}$ not to be injective, so there exists an open $V$ such that $$f^{\sharp}:O_Y(V) \rightarrow O_X(f^{-1}(V))$$ is not injective and ence look at the morphism induced between the fraction fields.
The equivalence $ii) \Leftrightarrow iii)$ from my point of view is just a consequence of definitions, so I think is easiest prove that $iii) \Rightarrow i)$.
Hint: If the image of $f$ is not dominant there exist an open subset of $Y$ such that $f^{-1}$ is empty...
