$h^0(S^nF(nD_1+D_2))=O(n^2)$ for a rank 2 vector bundle $F$ on a smooth curve

Question

The following proposition and proof are given in Lemma 2.5 of https://mathscinet.ams.org/mathscinet-getitem?mr=1272710, and I have some questions about it.

Proposition. Let $F$ be a rank 2 vector bundle on a smooth curve $C$, and $D_1,D_2$ divisors on $C$. Then $h^0(S^nF(nD_1+D_2))=O(n^2)$. (Here $n$ is a positive integer and $S^nF$ is the $n$-th symmetric product.)

Proof) Choose $D_3$ effective with $D_3\geq D_2$. Then $S^nF(nD_1+D_2)\subset S^nF(n(D_1+D_3))=S^n(F(D_1+D_3))$. So assume $D_1=D_2=0$, and $F$ very positive. Now use Riemann-Roch.

The proof is very short, but I cannot understand most of it.

1. How do we have $S^nF(nD_1+D_2)= (S^nF)\otimes O(nD_1+D_2)\subset (S^nF)\otimes O(n(D_1+D_3))= S^nF(n(D_1+D_3))$?

2. From 1, why can we assume $D_1=D_2=0$?

3. What is the definition of "very positive"? I cannot find it in the paper or google.

4. How does Riemann-Roch (https://en.wikipedia.org/wiki/Riemann%E2%80%93Roch_theorem#Statement_of_the_theorem) finish the proof?