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how can I show that $$\limsup_{n\to\infty} (a_n + b_n) \geq \limsup_{n\to\infty}(a_n) + \liminf_{n\to\infty}(b_n)$$

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See also this question: Properties of $\liminf$ and $\limsup$ of sum of sequences – Martin Sleziak Mar 25 '12 at 9:08
up vote 4 down vote accepted

Use $\limsup_n (x_n+y_n) \le \limsup_n x_n + \limsup_n y_n$, with $x_n = a_n+b_n$ and $y_n = -b_n$.

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Thank you very very much. – Yuri Mar 22 '12 at 2:32

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