I am thinking about a sequence such that the following holds (noticed > not $\geq$)
$\liminf_{n \to \infty} (a_n + b_n) > \liminf_{n \to \infty} a_n + \liminf_{n \to \infty} b_n$
I am not sure if this is allowed, but I tried doing something like
$a_n = \left \{2....-1,1,-1,1,-1,1 \right \}$
$b_n = \left \{2....1,-1,1,-1,1, -1\right \}$
I know the example doesn't work, but can you actually write down the limsup (the end term) like this? Or is this as erroneous as writing $1/0$?