$f(x)\in C_{ucb}$ (Uniformly continuous and bounded functions)
Show $cf(x)\in C_{ucb}$
Would it be enough to write:
$\forall \epsilon>0$; $ \exists \delta>0$ such that $|x-y|<\delta \rightarrow|cf(x)-cf(y)|<\epsilon^*$
$|f(x)-f(y)|<\frac{\epsilon^*}{|c|}$; where $c\neq0$ and $\frac{\epsilon^*}{|c|}=\epsilon$
I'm not using my bounded condition.