Given a double-indexed real sequence $\{ x_{n,m}\}$, do we have
$ \limsup_{n} \sum_m x_{n,m} \leq \sum_m \limsup_{n} \, x_{n,m}$
$ \liminf_{n} \sum_m x_{n,m} \geq \sum_m \liminf_{n} \,x_{n,m}?$
I am not sure about these, and just have some guess based on how sup and sum commute.
Thanks in advance!