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Few days ago I asked about the meaning of $$\limsup _{h\rightarrow \infty}\;\sup_{x\in \mathbb R}\; A(x,h)$$ being finite, and I got the answer. Now I'm facing another situation. Given that $A(x,h)$ and $B(x,h)$ are two positive functions in $x$ and $h$, is it true that

$$\limsup _{h\rightarrow \infty}\;\sup_{x\in \mathbb R}\; \bigg (A(x,h)B(x,h)\bigg )\leq \bigg (\limsup _{h\rightarrow \infty}\;\sup_{x\in \mathbb R}\;A(x,h)\bigg ) \; \bigg (\limsup _{h\rightarrow \infty}\;\sup_{x\in \mathbb R}\;B(x,h)\bigg ) $$

I feel it is correct! The same inequality is true for two bounded nonegative real sequences, as I know.

Note: It is known that both of the limits in the RHS exist and nonnegative.

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    Why the new account to ask this question?2011-08-31
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    The question Joan refers to is http://math.stackexchange.com/questions/60229/limit-supremum-finite-limit-meaning2011-08-31
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    @Joan: it would help if you consider registering your account. That way the system can better keep track of your questions and answers when you use different computers or when your IP address change. Thanks.2011-08-31

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