I am so rusty at this:

Theorem: Let f: A$\displaystyle \longrightarrow$B and g:B$\displaystyle \longrightarrow$C.

Prove that if f and g are onto, then g$\displaystyle \circ$f is onto.

Since g is onto $\displaystyle \forall$ c $\displaystyle \in$ C $\displaystyle \exists$ b $\displaystyle \in$ B such that g(b) = c.

Since f is onto $\displaystyle \forall$ b $\displaystyle \in$ B $\displaystyle \exists$ a $\displaystyle \in$ A such that f(a) = b.

Therefore $\displaystyle \forall$ c $\displaystyle \in$ C $\displaystyle \exists$ a $\displaystyle \in$ A such that g(f(a)) = c.