$f(x)g(x)$ is uniformly continuous if $f,g$ are uniformly continuous and $\lim_{x\to \infty}g(x)=0$?

Question

on $[0,\infty)$, can we prove $f(x)g(x)$ is uniformly continuous if $f,g$ are uniformly continuous and $\lim_{x\to \infty}g(x)=0$?

If $\lim_{x\to \infty}g(x)=0$ is droped, then it is easy to see that it is wrong with the example $f(x)=g(x)=x$.

Answer 1

Let $f(x)=x$ and $g(x)=\frac {\sin(x^{2})} x$ so that $f(x)g(x)=\sin(x^{2})$. $g$ is uniformly continuous because $g'$ is bounded. Also $g(x) \to 0$ as $x \to \infty$. Now let $x_n=\sqrt {n\pi}$ and $y_n=\sqrt {(n+\frac 12 ) \pi}$. Then $\sin (x_n^{2})-\sin (y_n)^{2}=0-1=-1$ for all $n$ even though $|x_n-y_n|=\frac {\frac 1 2 \pi} {x_n+y_n} \to 0$ as $n \to \infty$. Hence $fg$ is not uniformly continuous.