= Supremum of a sum
{title2=$\sup(A+B)=\sup A+\sup B$}
Let $A,B$ be nonempty <sets> of <real numbers> bounded above. Their <Minkowski sum> is $A+B=\{a+b:a\in A,\ b\in B\}$, allowing every pair from the <Cartesian product> $A\times B$. Then
$$
\boxed{\sup(A+B)=\sup A+\sup B.}
$$
The sum of the two <suprema> is an <upper bound> for every permitted sum. For the reverse bound, use the <approximation property of a supremum> to choose $a>\sup A-\varepsilon/2$ and $b>\sup B-\varepsilon/2$. Their sum is within $\varepsilon$ of the claimed value. This reasoning does not require either <supremum> to be a <maximum>.
Separate choices matter. On $X=[0,1]$, choose the <functions> $f(x)=x$ and $g(x)=1-x$. Each has <supremum> one, but their sum at a shared input is always one:
$$
\sup_{x\in X}(f(x)+g(x))=1<2=\sup_{x\in X}f(x)+\sup_{x\in X}g(x).
$$
The <maximum> of $f$ occurs at $x=1$, while that of $g$ occurs at $x=0$. A common input cannot realize both choices. With separate inputs, choosing $x=1$ and $y=0$ gives $f(x)+g(y)=2$. The allowed pairs of output values are the entire <unit square> for separate inputs, but only the <line segment> $a+b=1$ for the shared input.
\Image[/supremum-sums-separate-and-shared-inputs.png]
{title=Separate input choices fill a square; one shared input restricts the attainable sums}
{description=Original example $f(x)=x$, $g(x)=1-x$ on $[0,1]$. Left: every pair $(f(x),g(y))$ is permitted for separate inputs, including $(1,1)$ with sum two. Right: the shared-input pairs $(f(x),g(x))$ lie on $a+b=1$, with sum one. The colours encode $a+b$; this is a deterministic constraint on choices, not probabilistic <independence>.}
For any two <bounded functions> on the same nonempty <domain of a function>, the useful general estimate is
$$
\sup_x(f(x)+g(x))\leq\sup_x f(x)+\sup_x g(x).
$$
The <inequality> can be strict, as the example shows. Equality does not require attained <maxima>, but it does require that the two large values can be approached at compatible inputs. For example, $f(x)=g(x)=x$ on $(0,1)$ gives equality with value two, although neither <function> reaches its <supremum>. This distinction is useful when an <objective function> combines terms sharing a decision variable: adding the best possible termwise scores gives an <upper bound>, not necessarily a feasible combined score.
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