Prove that $$\limsup(a+b) \leq \limsup(a) + \limsup(b)$$ and $$\liminf(a+b) \geq \liminf(a) + \liminf(b)$$ where $a,b:\mathbb{N}\to \mathbb{R}$ are two bounded sequences.
I'm not sure where to begin on this proof.
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Martin Sleziak
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Uche Nwokelo
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You should begin by writing down the definitions of $\lim \sup (a+b)$, $\lim \sup(a)$, and $\lim \sup(b)$ – Alex G. Oct 13 '16 at 04:03
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This is very likely a duplicate. Have you checked the related questions? – Alexis Olson Oct 13 '16 at 04:15
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5See this post (and other posts linked there) and maybe also this post (and other posts linked there – Martin Sleziak Oct 13 '16 at 08:12
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Apart from Alexis Olson's suggestion to check related questions, another easy way to find posts about this inequality is to check question tagged limsup+inequality and then choose the frequent tab. – Martin Sleziak Oct 13 '16 at 08:14