Banach space and closed subspace

Question

Problem:

Banach space and closed subspace

Let I = [a,b] be a finite interval. Show that the space C(I,ℜⁿ) of continuous functions from I into ℜⁿ is a Banach space with the uniform norm.

                   llull∞ = sup lu(t)l

                              t∈I

(Show that this is a norm and that C(I,ℜⁿ) is complete).

(b) Let r > 0, x₀ ∈ ℜⁿ, and B = { x ∈ ℜⁿ : |x-x₀|< r }. Show that C(I,B) is a closed subspace of C(I,ℜⁿ), and is therefore a cpmpplete metric space.