Example closed sets for which $\inf(A+B) > \inf A+\inf B$

Question

A known inequality states:

$$\inf(A+B) ≥ \inf A+\inf B$$

Now what are example (closed) sets of the "$>$" case?

Answer 1

In that inequality $A$ and $B$ represent functions, defined on some domain (your question requests some closed interval).

Let $A(x) = \sin(2\pi x)$ and $B(x) = \cos(2\pi x)$ on the closed interval $[0,1]$. Then $$\inf(A) = \inf(B) = -1 \\ \inf(A) + \inf(B) = -2 \\ \inf(A+B) = -\sqrt{2} > \inf(A) + \inf(B) $$

Answer 2

Let $$ f, g:[0,1] \rightarrow \mathbb{R} $$ be given by $f(x) = x $ and $g(x) = -x$. We have that $sup \hspace{1mm} f = 1, sup \hspace{1mm}g = 0, sup\hspace{1mm}(f+g) = 0$, so $sup \hspace{1mm}f + sup\hspace{1mm} g = 1 > sup\hspace{1mm}(f+g) = 0$. Furthermore, we have $inf\hspace{1mm} f = 0, inf\hspace{1mm} g = -1, inf\hspace{1mm}(f+g) = 0$, and so $$inf \hspace{1mm}f + inf\hspace{1mm} g = -1<inf\hspace{1mm}(f+g) = 0$$