Let $X$ $\subset$ $\mathbb{R}$ and $f,g:X\rightarrow\mathbb{R}$ two continuous bounded functions. Is the product $fg$ a uniformly continuous function? Prove or give a counterexample.
I would say it is false, since $x\sin(x)$ isn't uniformly continuous and it could be defined in $X$=[-1,1] and still not being uniformly continuous. But I couldn't prove it was not uniformly continuous, for if I take two sequences, $\pi n$ and $\pi n + \pi/ n$, they will not be defined in the domain.
On the other hand, if I take let's say $\sin(x)$ and $\arctan(x)$, using the same sequences I get that it's difference is $0$ as $n\rightarrow\infty$. I know it's easy to prove that if the function were uniformly continuous, well that would be all (supposing they are bounded). But that isn't the case. Nor we can assume $X$ is compact, or the like. In summary I am stuck.
