# Isometric Isomorphism from $c_0$ to $C[0,1]$

I need to prove that there is an isomorphic isometry from $c_0$ to some subspace of $C[0,1]$. Researching a bit, it looks like it follows from Banach-Mazur theorem, but we haven't studied it, at least looks that way from my notes. Is there a "simple", direct way to prove this given that our domain is specifically $c_0$?

Let $x$ be in $c_0$. Define the function $f$ in $C[0,1]$ as follows: $f(1/n)=x_n$ for every $n$ and linear in between. The image of $c_0$ by this functional is clearly a subspace, and the norm is conserved.