$f$ and $g$ are bounded functions with common domain$D$
$\sup\limits_{x\in D}\big\{f(x)+g(x)\big\}\leqslant\sup\limits_{x\in D}f(x)+\sup\limits_{x\in D}g(x).$
Let both $S $and $T$ be non-empty subsets of $R$
$S+T=\begin{Bmatrix}z & \mid z=s+t,s\in S,t\in T\end{Bmatrix}.$ $\sup(S+T)=\sup S+\sup T$ What I want to know is, are these two issues related?
Note that $$\text{Im}(f + g) = \left\{f(x) + g(x) ~|~ x \in D\right\} \subseteq \text{Im}(f) + \text{Im}(g) = \left\{f(x) + g(y) ~|~ x,y \in D\right\}.$$ That's why you have an inequality \begin{align} \sup\limits_{x\in D}\big\{f(x)+g(x)\big\} &= \sup\text{Im}(f + g)\\ &\le \sup \left(\text{Im}(f) + \text{Im}(g)\right)\\ &= \sup \text{Im}(f) + \sup \text{Im}(g)\\ & = \sup\limits_{x\in D}f(x)+\sup\limits_{x\in D}g(x) \end{align}