# Liminf/Limsup of a Sum of Sequences

• November 4th 2010, 10:06 PM
valerian
Liminf/Limsup of a Sum of Sequences
$(a_n)$ and $(b_n)$ are bounded sequences.

Prove the following:
$\liminf a_n + \liminf b_n \leq \liminf (a_n + b_n)$
$\leq \limsup a_n + \liminf b_n \leq \limsup (a_n + b_n)$

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 (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.
• November 4th 2010, 10:50 PM
Drexel28
Quote:

Originally Posted by valerian
$(a_n)$ and $(b_n)$ are bounded sequences.

Prove the following:
$\liminf a_n + \liminf b_n \leq \liminf (a_n + b_n)$
$\leq \limsup a_n + \liminf b_n \leq \limsup (a_n + b_n)$

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 (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.

How do you define $\limsup$, etc.? The easiest definition for this one is that it's the supremum of the set of all subsequential limits of the sequence.
• November 4th 2010, 11:00 PM
valerian
Yes, that's the definition I'm working with. Liminf is the infinum of all subsequential limits likewise. I'm having trouble working with the terms that are the liminf or limsup of a sum of the sequences.