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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](http://math.stackexchange.com/questions/70478/)2012-03-25

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