A proof in real analysis If we let $f$ and $g$ be Riemann integrable functions on $[a,b]$ and $c\in \mathbb{R}$ be a constant.
I need to show that $$\displaystyle \int_a^b cf(x)\,dx=c\int_a^b f(x)\,dx.$$
My idea here was to consider both cases, when $c>0$ and when $c<0$.
Case 1: 
Suppose $c>0$. Let $P$ be a partition.
Then $L(cf, P) = c L(f,P)$ and $U(cf,P) = c U(f,P)$.
Thus,
$$\int^b_{a} cf(x)\,dx = c \int^b_a f(x)\,dx.$$
Case 2:
Suppose $c<0$. 
Then $L(cf,P) = cU(f,P) $ and $U(cf,P) = cL(f,P) $.
Thus,
$$\int^b_a cf(x)\,dx = c \int^b_a f(x)\,dx.$$
Is this proof sufficient? Or do I need to consider another way to prove this?

Your proof is fine. You'd be having the same results, had you used Riemann sums.
