Ring of global sections of a large twist of a vector bundle is not zero This is Vakil 18.5 D (the first claim), self-study. Please note this is an intermediate step in the classification of vector bundles on $\mathbb P^1_k$, so we may not use that result.
We have to show that if $\mathcal E$ is a rank $r$ locally free sheaf on $\mathbb P^1_k$, then for $m <<0$,
$$\operatorname{Hom}(\mathcal O(m), \mathcal E) \neq 0$$
To begin, we note that the sheaf Hom $\operatorname{\mathcal Hom}(\mathcal O(m), \mathcal E) \simeq \mathcal E(-m)$ by a previous exercise. Hence we need to show
$$\Gamma (\mathbb P^1_k, \mathcal E(-m)) \neq 0$$
for $m << 0$. Vakil then says to use Serre vanishing, which gives us that
$$H^i (\mathbb P^1_k, \mathcal E(-m)) = 0$$
for $m <<0$ and $i > 0$, but I do not see why this forces $H^0$ to be nonzero.
 A: Attempt at a solution:
We wish to show $\Gamma (\mathbb P^1_k, \mathcal E(-m)) \neq 0$ for $m <<0$.
Filter $\mathcal E$ as
$$0 = \mathcal E_0 \subset \mathcal E_1 \subset ... \subset \mathcal E_r = \mathcal E$$
such that each subquotient is some invertible sheaf $\mathcal L_i \simeq \mathcal O(n_i)$ on $\mathbb P^1_k$ using Stacks OAYP (which does not use the classification of vector bundles on $\mathbb P^1_k$, and I have been through its proof before for previous exercises.)
This admits an exact sequence
$$0 \to \mathcal E_{r - 1} \to \mathcal E \to \mathcal O(n_r) \to 0$$
By Serre Vanishing, if $m<<0$, $H^i(\mathbb P^1_k, \mathcal E(-m)) = 0$ for all $i > 0$.
Tensor the above exact sequence by $\mathcal O(-m)$ for $m << 0$ that the above holds; another condition will be added to the size of $m$ momentarily. This preserves exactness, and gives
$$0 \to \mathcal E_{r - 1}(-m) \to \mathcal E(-m) \to \mathcal O(n_r - m) \to 0$$
By the Serre vanishing observation above, taking global sections is an exact functor, giving
$$0 \to \Gamma (\mathbb P^1_k, \mathcal E_{r - 1}(-m)) \to \Gamma(\mathbb P^1_k,\mathcal E(-m)) \to \Gamma(\mathbb P^1_k,\mathcal O(n_r - m) ) \to 0$$
But we already know by Cohomology of $\mathbb P^n_A$ that $\Gamma(\mathbb P^1_k,\mathcal O(n_r - m) ) \neq 0$ for $m << 0$, so impose that additional condition on the size of $m$ before you twist above, and you will have
$$\Gamma(\mathbb P^1_k,\mathcal E(-m)) \neq 0$$
