Define the space $\beta^1([a,b])$ as the space of functions $f : [a,b] \mapsto \Bbb R$ which are everywhere differentiable and whose derivative $f'$ is a bounded function. One equips this space with the metric $$ d(f,g) = \sup{|f(x)-g(x)|} + \sup{|f'(x) - g'(x)|}$$ Prove that this turns $\beta^1([a,b])$ into a complete metric space.
Please help. I know what a complete metric space is, but I'm having trouble getting started with this proof. Is there a way to do this proof without knowing about Banach spaces? I am stuck and any help would be appreciated.
Thanks