If $f: [a,b] \rightarrow \Bbb R$ is bounded and $c \in \Bbb R_+$. Show that for any partition $\{x_o,...,x_n\}$ of $[a,b]$ that $U(P,cf)= cU(P,f)$. Suppose $f: [a,b] \rightarrow \Bbb R_+$ is bounded and $c \in \Bbb R$. Show that for any partition $\{x_o,...,x_n\}$ of $[a,b]$ that $U(P,cf)= cU(P,f)$.
Edit: My notation $U(P,f)$ is an upper darboux sum on a closed interval for the function $f$ with partition $P$.
So what I understand is that I am to show that
$\sum_{i=0}^{n} \sup \{cf(x)| x \in [x_{i-1},x_i] \} (x_i-x_{i-1}) $ is equal to 
$c \cdot\sum_{i=0}^{n} \sup \{f(x)| x \in [x_{i-1},x_i] \} (x_i-x_{i-1})$
Edit: Originally the problem was stated with $c \in \Bbb R$ but this had problems, so that it only makes sense if $c \geq 0$
 A: A stronger claim can be proven if $f$ has a Riemann integral, yet the process for the proof is the same.
Suppose you have two functions $f(x)$ and $g(x)$ such that $g(x)=f(x/c)$ for some constant $c$, and $f$ has a Riemann integral.
Then prove:
$$\int_{ac}^{bc}g=c\int_a^bf$$
Construct two partitions $P_0, \ Q_0$ such that:
$$P_0=\left\{x_0,x_1,...,x_n\right\} \ on \ [a,b]\ and \ Q_0=\left\{cx_0,cx_1,...,cx_n\right\} \ on \ [ac,bc]$$
If we define:
$$m_i=inf\left\{f(x):x\in[x_{i-1},x_i]\right\} \ and \ M_i=sup\left\{f(x):x\in[x_{i-1},x_i]\right\} \ for \ i=1,2,...,n$$
Then the lower sums evaluate as follows:
$$L(P_0,f)=\sum_{k-1}^nm_i(x_i-x_{i-1}) \ and \ L(Q_0,g)=c\sum_{k-1}^nm_i(x_i-x_{i-1})=cL(P_0,f)$$
Likewise, the upper sums are as follows:
$$U(P_0,f)=\sum_{k-1}^NM_i(x_i-x_{i-1}) \ and \ U(Q_0,g)=c\sum_{k-1}^nM_i(x_i-x_{i-1})=cU(P_0,f)$$
It follows directly from the definition of a Riemann integral, that the claim holds, that is:
$$\int_{ac}^{bc}g=c\int_a^bf$$
A: Using the fact that if $c \geq 0$ then $c \cdot sup(A) = sup (cA)$ the conclusion follows.
