I know that for any two real sequences x_n and y_n, we have(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\liminf_{n\to \infty} x_n + \liminf_{n\to \infty} y_n \leq \liminf_{n\to \infty} (x_n + y_n).

[/tex]

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

[tex]

\liminf_{n\to \infty} x_n + \liminf_{n\to \infty} y_n = \liminf_{n\to \infty}(x_n + y_n),

[/tex]

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!

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# Question about liminf of the sum of two sequences

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