I'm to show that the product of two uniformly continuous functions on $\Bbb R$ don't have to be uniformly continuous. I know that the product is iff the two functions are bounded... I'm assuming it would be enough to give an example of uniformly continuous unbounded functions which, when multiplied together, are not uniformly continuous.
As I know $x^2$ is NOT uniformly continuous on $\Bbb R$, would it be enough to say that if $f(x) = g(x) = x$ then $(fg)(x) = x^2$ shows the product need not be uniformly continuous?