Let $(a,b,c,d)$ be a 4-tuple of real numbers such that $a \leq b \leq c \leq d$, but is otherwise arbitrary. Are there real sequences $x\_n$ and $y\_n$, such that lim inf $x\_n$ $+$ lim inf $y\_n$ $= a$, lim inf $(x\_n + y\_n) = b$, lim sup $(x\_n + y\_n) = c$, and lim sup $x\_n$ $+$ lim sup $y\_n$ $= d$?



Can this string of inequalities take on arbitrary values?
No. If $a=0$, then we may add a constant to $x\_i$ and subtract it from $y\_i$ to achieve $\liminf x\_n=\liminf y\_n=0$; this does not affect $b$, $c$, and $d$. Without loss of generality, assume that $\limsup x\_n\geq d/2$; then $c\geq d/2$ as well.


Similarly, since $\limsup y\_n\leq d/2$, we get $b\leq d/2$.


Conversely, it seems that any 4-tuple with $b\leq (a+d)/2\leq c$ is reachable. Set $\liminf x\_n=\liminf y\_n=a/2$, $\limsup x\_n=\limsup y\_n=d/2$, achieved on sparse pairwise disjoint sequences of indices. Now it is easy to reach $b$ and $c$.


