Is the composition of two injective functions injective?

Question

Consider $f:A \to B$ and $g: B \to C$. If I know that both $f$ and $g$ are injective, then can I state that $$g \circ f:A \to C$$ is injective?

Answer 1

Yes, $$f(a) = f(b) \iff a =b,$$ $$g(a) = g(b) \iff a =b,$$ so

$$g(f(a)) = g(f(b)) \implies f(a) = f(b) \implies a = b. $$

Answer 2

Using "injective iff has a left inverse":

$f$ injective $\implies$ $\exists\tilde f:\ \tilde f\circ f = I_A$.

$g$ injective $\implies$ $\exists\tilde g:\ \tilde g\circ g = I_B$.

Now, the composition has left inverse, namely $\tilde f\circ \tilde g$: $$(\tilde f\circ\tilde g)\circ(g\circ f) = \tilde f\circ\tilde g\circ g\circ f = \tilde f\circ I_B\circ f = \tilde f\circ f = I_A.$$