Question about liminf of the sum of two sequences

[AxiomOfChoice]

I know that for any two real sequences x_n and y_n, we have

<br /> \liminf_{n\to \infty} x_n + \liminf_{n\to \infty} y_n \leq \liminf_{n\to \infty} (x_n + y_n).<br />

I also know that, if one of the sequences converges, the inequality becomes equality. My question is this: If I've managed to show that

<br /> \liminf_{n\to \infty} x_n + \liminf_{n\to \infty} y_n = \liminf_{n\to \infty}(x_n + y_n),<br />

can I conclude that one, or both, of the sequences converge? A simple yes/no would suffice, but (of course) I'd prefer a short proof or counterexample. Thanks!

 

[statdad]

What if

<br /> x_n = y_n = (-1)^n<br />

 

[AxiomOfChoice]

What if

<br /> x_n = y_n = (-1)^n<br />

Lame! I was hoping both of the sequences had to converge! And what a simple counterexample to prove me wrong! Thanks, though