Liminf/Limsup of a Sum of Sequences

and are bounded sequences.

Prove the following:

I understand why the second term is greater or equal to the first term but I am having trouble proving mathematically. Mentally, I recognize that they can be equal only when the "liminf terms" of each sequence "line up" (I know this is terrible math terminology (Lipssealed)) with each other so they are equivalent to the liminf of the sum. I see similarities between this and the triangle inequality but I can't find a way to apply it.

As for the other terms in the inequality, I haven't looked at them much yet as I've been hung up on this one. Any thoughts or help with this would be greatly appreciated.