Prove that $f \circ g$ is injective if $f$ and $g$ are injective.

Question

Is this proof correct?

Proof:

Let $g:A \rightarrow B$, $f:B \rightarrow C$ and $x,y\in A$.

$x=y \iff g(x)=g(y) \iff f(g(x))=f(g(y)) \iff f\circ g(x)=f \circ g(y)$.

We have proved $f\circ g(x)=f \circ g(y) \Rightarrow x=y$.

Hence, if $f$ and $g$ are injective, so is $f \circ g$.

Answer 1

This is completely correct. Well-done!

Follow-up: show that the same holds when 'injective' is replaced by 'surjective'.