$\displaystyle (a_n)$ and $\displaystyle (b_n)$ are bounded sequences.

Prove the following:

$\displaystyle \liminf a_n + \liminf b_n \leq \liminf (a_n + b_n)$

$\displaystyle \leq \limsup a_n + \liminf b_n \leq \limsup (a_n + b_n)$

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

) 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.