A proof in real analysis

Question

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?

Answer 1

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

Answer 2

Ya, I think if you were going to be precise about it, you'd do $L \leq \int f(x) \leq U$, then $\int cf(x)$ is between cU and cL. The order of the inequality would be why you need two cases.