If $f\in R(\alpha)$ and $C\in\mathbb{R}$, then $Cf\in R(\alpha)$ and $\int_a^b Cf\operatorname{d}\alpha=C\int_a^b f\operatorname{d}\alpha.$

Question

I can figure out how to prove this, assuming C is positive, but I'm not sure how to take into consideration if C < 0.

The sup of any partition of $Cf$ will simply be Csupf(x). This makes the proof for C > 0 easy, but I'm not sure what the differences need to be for C < 0.