For a pure $d$-dimensional sheaf $E$ on a projective algebraic variety over a field $k$, one has the Harder-Narasimhan filtration $$0\subset E_1\subset E_2\subset...\subset E_{l-1}\subset E_l:=E,$$ where the successive quotients $E_i/E_{i-1}$ are semistable with reduced Hilbert polynomials $p_i$ satisfying $p_{max}:=p_1>p_2>...>p_{l-1}>p_l:=p_{min}$. From the construction and uniqueness of the HN filtration, we know that $E_1$ is the maximal destabilizing sub sheaf of $E$, i.e. the unique subsheaf satisfying $p(E_1)\geq p(G)$ for all subsheaves $G\subset E$ such that $p(E_1)=p(G)$ implies $G\subset E_1$. The quotient $E/E_{l-1}$ is known as the minimal destabilizing quotient.

For some silly reason I'm having trouble showing that if $E\rightarrow G$ is any nontrivial quotient of $E$ then $p(G)\geq p_{min}$. Can anyone help me with this?

Also is there an intrinsic definition of the minimal destabilizing quotient?