Prove function is strictly decreasing

I've got functions $f,g:\mathbb{R}\rightarrow \mathbb{R}$ . $f$ is strictly increasing and $f\circ g$ is strictly decreasing I am asked to show $g$ is strictly decreasing. I already have a solution in front of me however I would like to know if this approach is correct.

Suppose $x_{1} < x_{2}$ then $f\left ( g\left ( x_{1} \right ) \right ) > f\left ( g\left ( x_{2} \right ) \right )$ $\Rightarrow g\left ( x_{1} \right ) > g\left ( x_{2} \right )$ since $f$ is strictly increasing.

therefore $g$ is strictly decreasing.