Ampleness and Extensions of Line-Bundles I believe I have a proof that any vector bundle $V$ of rank $n$ on a projective variety $X$ has a filtration by line bundles (that is, there is a filtration $V = L_n \subset ... \subset L_0$, where $L_{i+1}/L_{i}$ is a line bundle for all $i$). This seems a little bit too good to be true. Could anybody verify my proof, or point out where the mistake is?

Let $H$ be an ample line bundle on $X$. Then for some $p$, $H^p \otimes V$ has a global section, corresponding to an injection $\mathscr{O_x} \to H^p \otimes V$. This induces a exact sequence $0\to H^{-p} \to V \to M \to 0$, for a vector bundle M of rank $n-1$. By induction, $M$ has a filtration by line bundles. Hence, adding $H^{-p}$ to this filtration, so does $V$. 

 A: This is not true.
First a counterexample: let $X=\mathbf P^2$ and $T$ the tangent bundle. I claim there is no exact sequence
$$ 0 \rightarrow L_1 \rightarrow T \rightarrow L_2 \rightarrow 0$$
where $L_1$ and $L_2$ are line bundles. To see this, note that by the Euler sequence, the total Chern class of $T$ is $c(T)=1+3H+3H^2$ where $H$ is the hyperplane class. It is then easy to check that $c(T)$ does not factor as $(1+aH)(1+bH)$ for any integers $a$ and $b$. 
Why does your proof not work? It is because the injection you (correctly) find is only an injection in the category of sheaves. Such an injection need not induce injections on all fibres of the bundles in question, so the quotient need not be a vector bundle. For a more extreme illustration of the same idea, if $L$ is any line bundle with a global section, we get an injection $\mathcal O \rightarrow L$. But (unless $L$ is trivial) this map will be zero on some fibres, and the quotient will be a non-trivial torsion sheaf. The moral of the story is that vector bundles do not form a good category, which is part of the reason for working with sheaves instead.
Finally, let me mention a weaker result that is true: the existence of Harder--Narasimhan filtrations. I'll let you search for the details, but the idea is that we can find a filtration of any vector bundle whose quotients are semistable vector bundles, hence still "simple" in some sense.
