Define the space *β*1([*a*,*b*])

as the space of functions *f*:[*a*,*b*]↦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 *β*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. I would prefer a hint that would point me in the right direction as I'd like the opportunity to work through the proof myself, but I'll be thankful for any help I receive.

Thanks