I'm trying to prove:

If $\displaystyle f$:$\displaystyle A --> R$ and $\displaystyle g$:$\displaystyle A --> R$ are uniformly continuous functions, then $\displaystyle f * g$ is uniformly continuous.

I've gotten to the fact that

|f(x)g(x) - f(y)g(y)| <= |f(x)| * |g(x)-g(y)| + |g(y)| * |f(x) - f(y)|

But I'm not sure where to go from here...