So I'm going through class notes to try and prepare for an analysis qual. There are a number of little proofs in which the professor gave either no details or a brief sketch. I wanted to check if this was the right approach. We have by definition that
$\int_E cf(x) = \sup _{h(x) \le cf(x)} \{\int_E h(x)\}$
Now I want to say that this expression is equal to
$\sup _{g(x) \le f(x)} \{\int_E cg(x)\}$
It seems like this follows by just noting that for any simple function $h$ then $g(x) = c h(x)$ is a simple function and $\int_E h(x) = \int_E c g(x)$. Then we have
$\int_E cf(x) = \sup _{g(x) \le f(x)} \{c \int_E g(x)\} = c \sup _{g(x) \le f(x)} \{\int_E g(x)\} = c \int_E f$
Note, I'm assuming that $\int c f = c \int f$ for simple functions f has always been established. Hopefully this isn't too annoying to read through. Is this correct? Thanks for your time!