How would one show that the set consisting of the monomials $1,x,x^2,...$ is a closed subset of the metric space $C[a,b]$ under the metric $d(a,b) = ||a-b|| =sup_{[0,1]}|a-b|$ ?
I considered its complement and had a hard time getting started; since given any element which is a sum of these terms (possibly with different coefficients, I don't see where to start in showing that an $\epsilon$-Ball of that element would be contained in the complement. Proving every convergent sequence in my set of monomials has a limit in there is an option, but I really need some guidance on how to set up that kind of proof since I don't feel clever enough yet to pull of a proof with sequences.