# proving an inequality involving limit superior and limit inferior

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

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