Why $\sup (f+g) \leq \sup\ f + \sup\ g$, where f & g are functions? while sup (A + B) = sup A + sup B, where A and B are sets?
Could anyone explain this for me please?
2026-04-05 19:05:25.1775415925
Supremum of a set and Supremum of a function.
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When one writes $A+B$ where $A$ and $B$ are sets, one typically means the set containing all of the sums that can be formed by adding one element from $A$ and one element from $B$. Therefore, the supremum of $A+B$ is the sum of the two supremums.
On the other hand, when one adds two functions together, the two functions are only added pointwise. Therefore, intuitively, if $f$ attains its maximum $y$ at a point $x$ and $g$ attains its maximum $y'$ at a point $x'\neq x$, there is a priori no reason why $f+g$ should attain $y+y'$.