$(C^1[0,1], || . ||_\infty)$ is not Banach Space.

Question

Recently, I am studying Banach Space. I know that $(C[0,1], || . ||_\infty)$ is a Banach Space and closed subspace of Banach Space is Banach. This follows from the result that closed subset of complete space is complete. I have shown that $(C^1[0,1], || . ||_\infty)$ is not a closed subspace of $(C[0,1], || . ||_\infty).$ This can be shown using "Stone Weierstrass" theorem. Up to this is fine. Now my problem is from here can I say that $(C^1[0,1], || . ||_\infty)$ is not Banach? Or I have to give a counter-example so that $(C^1[0,1], || . ||_\infty)$ is not complete. Please help me. Thanks in advance.

Answer 1

What about a sequence of $\mathcal C^1(\mathbb R)$ function that converges uniformly to $|x|$ ?

Answer 2

What about the sequence $(x_n)\subset C^1[0,1]$ given by $x_n=x^n$?