Polynomials in $y=e^x$ dense in C[1,e]

Question

I try to understand a proof of Theorem that I found in Tran van Thuong article. \ Theorem: For each integer $N\ge1$, the set of functions: {$e^{nx} : n \ge N$} has a linear span dense in $C[0,1]$. Proof: Put $y=e^x$. By Weierstrass Approximation Theorem, polynomials in y are dense in $C[1,e]$. Moreover given any $f ∈ C[1,e]$, and any $\epsilon>0$ there is a finite sum: $A_Ny^N + A_{N+1}y^{N+1} + \cdots + A_{N+L}y^{N+L} = Q(y) $, such that $|f(y)-Q(y)|<\epsilon$. \ My question is: Why polynomials in y are dense in $C[1,e]$ ? And why we consider $f ∈ C[1,e]$ ?