# How to prove these inequalities: $\liminf(a_n + b_n) \leq \liminf(a_n) + \limsup(b_n) \leq \limsup(a_n + b_n)$

• Prove $\limsup\limits_{n \to \infty} (a_n+b_n) \le \limsup\limits_{n \to \infty} a_n + \limsup\limits_{n \to \infty} b_n$
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Hint: First show that $\limsup$ is subadditive, that is,
$\limsup(a_n + b_n) \le \limsup a_n + \limsup b_n$
for real sequences $(a_n)$, $(b_n)$. From this conclude using $-\limsup(-a_n) = \liminf a_n$ that $\liminf$ is superadditive (inequality $\ge$ in the above). Then you can use all this to prove
$\liminf (a_n + b_n) – \limsup b_n = \liminf (a_n + b_n) + \liminf(-b_n) \le \liminf a_n$