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.