On the space of continuous functions on $[0,1]$, I have a metric
$$d(f,g) = \sup | \alpha(x) (f(x) -g(x))|,$$
where $\alpha(x)$ is a continuous function and $\alpha(x) \ne 0$.
I'm trying to find whether this space is complete. I think it is complete because the continuous functions on $[0,1]$ are complete and the $\alpha$ doesn't create a problem because it is continuos on a closed interval and so bounded. But this seems a bit too simple?