When is the infimum of the sum of two sets equal to the sum of their infima? When is the following true? $A$ and $B$ are subsets of real numbers. I don't say that $A$ and/or $B$
are closed:
$$\inf (A + B) = \inf (A) + \inf(B)$$
When is there a strict inequality in between? Could you give an example?
When is the equality for what sufficient condition? Could you give an example? 
Thanks a lot!
 A: First, let $a=\inf A$, $b=\inf B$. Pick $x+y\in A+B$. Then $$a\leq x,b\leq y\implies a+b\leq x+y$$
Second, let $\epsilon >0$ be given. By definition of the infimum, there exists $x$ in $A$ and $y$ in $B$ such that $$x<\inf A+\frac \epsilon 2$$ $$y<\inf B+\frac \epsilon 2$$
By summing $$x+y<\inf A+\inf B+\epsilon$$
The first step shows $$\forall z\in A+B\text{ we have }\inf A+\inf B\leq z$$
The second step shows $$\forall \epsilon >0\;\exists z\in A+B \text{ such that }z<\inf A+\inf B+\epsilon$$
This is precisely the (equivalent) definition of $\inf(A+B)$.
A: First notice that the infimum of a set $X$ exists and it's finite if and only if $X$ is non empty set bounded below so let's assume that $A$ and $B$ have these conditions.
As you said the inequality $$\inf(A)+\inf(B)\leq \inf(A+B)$$
is pretty clear so let's prove the other inequality:
We have $$\inf(A+B)-b\leq a,\, \forall a\in A, b\in B$$
so 
$$\inf(A+B)-b\leq \inf(A),\, \forall  b\in B\iff \inf(A+B)-\inf(A)\leq b,\, \forall  b\in B$$
and then
$$\inf(A+B)-\inf(A)\leq \inf(B)\iff \inf(A+B)\leq\inf(A)+ \inf(B)$$
