Small doubt in understanding the proof of showing C[0,1] is Banach space

Question

I am reading Functional analysis by M. Fabian. Author gave the proof that C[0,1] is Banach space and then says that we can generalise the result for any compact set K. And now after reading this generalise statement I am wondering where he has actually use the closeness and boundedness of [0,1]. Like if I take (0,1) where will proof go wrong?( The proof I guess will be similar in all texts).

Answer 1

In $[0,1]$ (or in a general compact space $K$), the distance that we are using is$$d(f,g)=\sup_{x\in[0,1]}\bigl\lvert f(x)-g(x)\bigr\rvert.$$In, say, $(0,1)$ this metric doesn't make sense, because then there are unbounded continuous functions and therefore that $\sup$ may not exist.