Let $f, g: Q\rightarrow \mathbb{R}$ be bounded functions such that $f(x)\leqslant g(x)$ for $x \in Q$.

Question

Let $f,g:Q\to\mathbb R$ be bounded functions such that $f(\mathbf x)\le g(\mathbf x)$ for $\mathbf x\in Q$. Show that $⨜_Q f \le ⨜_Q g$ and $⨛_Q f \le ⨛_Q g$.

I tried the following way: as $m_ R=\inf \{f(x):x\in R\}$ and $m'_ R=\inf\{g(x):x\in R\}$ so $m_R\leqslant m'_R$ because $f(x)\leqslant g(x)$ for $x \in Q$ so $m_RV(R)\leqslant m'_RV(R)$ so $\sum_R m_RV(R)\leqslant \sum_R m'_RV(R)$ then $L(f,P)\leqslant L(g,P)$ so $\sup_P\{L(f,P)\}\leqslant \sup_P\{L(g,P)\}$ so $\underline\int_{Q}f\leq \underline\int_{Q}g$ is well?