Approximation of $L^\infty$ functions by $\mathcal{D}$ functions?

Question

Let $Q=(0,T)\times\Omega$ with $\Omega$ a bounded domain. I read this:

"the inequality holds for all $f \in \mathcal{D}(Q)$, and by approximation, it holds for all $f \in L^\infty(Q)$ since the constants only depends on the $L^\infty$ norm of $f$."

What approximation is this? I thought it was density, but no such density result holds.

Edit The inequality is $$\int_Q (u_1-u_2)f \leq C_{f}\times C_{u_1,u_2}$$ where $C_f$ is the constant that depends only on the $L^\infty$ norm of $f$. Here $u_i$ are $L^\infty(Q)$ functions.

Answer 1

Given $f \in L^\infty$, you can find a sequence $f_n \in C_c^\infty$ such that $\|f_n\|_\infty \le \|f\|_\infty$, and $f_n \to f$ in $L^1$, that is, $\|f-f_n\|_1 \to 0$.