Assume $F: X\rightarrow Y$ and $G: Y \rightarrow Z$ are surjective prove that $G \circ F$ is surjective.
Let $z\in Z$ Set x$\in X$ to be such that $F(x)=y$ and $G(y)=z$ (these exist because F and G are both surjective). Hence $(G \circ F)(x)$ $=G(F(x)) =G(y)=z$.
Is this proof correct?