1. ## uniform continuity problems

I need help solving the following:
Assume f and g are uniformly continuous on D.

First I need to prove that if f and g are both bounded on D, that fg is uniformly continuous on D.

I also need to prove that if D is closed and bounded, then fg is uniformly continuous.

Also, if $\displaystyle g(x)\neq 0$ for any x in D and D closed and bounded, then $\displaystyle \frac{f}{g}$ is uniformly continuous.
Any help on these is greatly appreciated thanks!

2. Originally Posted by dannyboycurtis
I need help solving the following:
Assume f and g are uniformly continuous on D.

First I need to prove that if f and g are both bounded on D, that fg is uniformly continuous on D.

I also need to prove that if D is closed and bounded, then fg is uniformly continuous.

Also, if $\displaystyle g(x)\neq 0$ for any x in D and D closed and bounded, then $\displaystyle \frac{f}{g}$ is uniformly continuous.
Any help on these is greatly appreciated thanks!
$\displaystyle |f(x)g(x)-f(y)g(y)|\leq|f(x)||g(x)-g(y)|+|g(y)||f(x)-f(y)|$