Let $F\rightarrow \mathbb{P}^{1}$ be a vector bundle, and let $\mathcal{F}$ be its sheaf of sections. I have read that the sheaf $\mathcal{F}(m)/\mathcal{F}$ is a skyscraper sheaf, but I do not know how to prove it.
I do not know if it is of any help, but I am going to show what I have thought until now. We have $$ \mathcal{F}(m)=\mathcal{F}\otimes\mathcal{O}_{\mathbb{P}^{1}}(m). $$ Since $H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(m))\simeq K[X_{0},X_{1}]_{m}$, I guess that we may consider $(X_{1})$ as the distinguished point of $\mathbb{P}^{1}$ with respect to $\mathcal{F}(m)/\mathcal{F}$. Nevertheless, I do not know how to continue.
Any help would be appreciated.
This works for any curve $C$. If $p$ a point on $C$ there is a short exact sequence $$ 0 \to \mathcal{O}(-p) \to \mathcal{O} \to k(p) \to 0. $$ Take a tensor product of this exact sequence with $\mathcal{F}(p)$. It stays exact because $F$ is a vector bundle. So, you get a short exact sequence $$ 0 \to \mathcal{F} \to \mathcal{F}(p) \to \mathcal{F}(p)\otimes k(p) \to 0. $$
Sheaf $\mathcal{F}(p)\otimes k(p) \cong \mathcal{F} \otimes k(p)$ is a skyscraper sheaf supported on $p$.
On $\mathbb{P}^1$ we have $\mathcal{O}(1) = \mathcal{O}(p)$ and $\mathcal{O}(m) = \mathcal{O}(mp)$, where $m \in \mathbb{Z}$. For positive $M$ we generalize previous argument by starting with short exact sequence $$ 0 \to \mathcal{O}(-mp) \to \mathcal{O} \to k_m(p) \to 0. $$ where $k_m(p)$ is a skyscraper sheaf supported on $x$ of length $m$. Then we tensor this short exact sequence with vector bundle $\mathcal{F}(mp)$.