Left Inverse Mapping is Surjection

Theorem
Let $f: S \to T$ be a mapping.

Let $g: T \to S$ be a left inverse of $f$.

Then $g$ is a surjection.

Proof
By the definition of left inverse:


 * $g \circ f = I_S$

where $I_S$ is the identity mapping on $S$.

$I_S$ is a surjection by Identity Mapping is Surjection.

Thus $g \circ f$ is a surjection, and by Surjection if Composite is Surjection we see that $g$ is also a surjection.