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$$ \limsup \left(f(h)+g(h)\right) \leq \limsup f(h)+ \limsup g(h).$$

How can we prove this? Any help would be appreciated.

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    $ h \in $ what?2011-10-03
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    Thomas, it is very rare for a thread on math.stackexchange to have no activity for 2 hours, plenty of us are willing to help you as long as you show us some effort and work you have done!2011-10-03
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    To expand on Mark's comment: What is $h$ in your formula? Is this a sequence, so that $h=1,2,3,\ldots$?2011-10-03
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    In general, to prove that lim sup (whatever) is less than or equal to (something), it suffices to prove that for every $\epsilon > 0$, eventually (whatever) is less than (something + $\epsilon$). Here "eventually" means when $h$ is sufficiently large, if you want $h\to\infty$, or else $|h-a|$ is sufficiently small, if you want $h\to a$.2011-10-03

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