AxiomOfChoice
Oct22-09, 11:11 AM
I know that for any two real sequences x_n and y_n, we have
\liminf_{n\to \infty} x_n + \liminf_{n\to \infty} y_n \leq \liminf_{n\to \infty} (x_n + y_n).
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
\liminf_{n\to \infty} x_n + \liminf_{n\to \infty} y_n = \liminf_{n\to \infty}(x_n + y_n),
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!
\liminf_{n\to \infty} x_n + \liminf_{n\to \infty} y_n \leq \liminf_{n\to \infty} (x_n + y_n).
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
\liminf_{n\to \infty} x_n + \liminf_{n\to \infty} y_n = \liminf_{n\to \infty}(x_n + y_n),
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!