The Composition Of Surjective Functions Is Surjective Take 2

First note that a function is surjective if and only if the image of the domain equals the codomain. We will use both the "if" and "only if" parts of that statment. We now begin the proof. Because the two functions are surjective, we know $f(A)=B$ and $g(B)=C$ (we used the "only if", or implies, part). This means that $(g\circ f)(A)=g(f(A))=g(B)=C$. Since we know $(g\circ f)(A)=C$, this proves that $g\circ f$ is surjective (we used the "if" part).