Suppose that $g \circ f$ is the composition of $f:X \to Y$ and $g:Y \to Z$. Suppose that $g \circ f$ is onto.
- Prove that if $g$ is one-to-one, then $g$ is bijective and $f$ is onto.
This is the last part of a series of problems stemming from the original statement; I proved that g is surjective, but this part of the problem has me stumped. Maybe it's something really obvious that I'm missing, but I generally don't have a good grasp on injections, surjections, and bijections, so maybe not.