Proof that in the Banach space $C^0[0,1]$ the set of functions in $C^1[0,1]$ is a meagre set How can I prove that in the Banach space $C^0[0,1]$ the set of functions in $C^1[0,1]$ is a meagre set? How do I go about proving this statement? I can't find any direct union of nowhere dense set I could write it as, nor an indirect trick.

hint: You may have a look at the sets: $E_n = \\{ \phi \in C^1([0,1]): \sup |\phi'|\leq n \\}$, $n\geq 1$