Prove $ C^1[0,1]$ is not a Banach space with the sup-norm

Question

Let $ C^1[0,1]$ be the normed space of continuously differentable functions on $[0,1],$ with $||x||=\max_{t\in [0,1]} |x(t)| $. Prove that $ (C^1[0,1],||.||)$ isn't a Banach space

I think we need to construct a Cauchy sequence in the space which doesn't converge in the space but i don't know how to construct this sequence. I'm stuck here

Answer 1

The functions $f_n(x) = ((x-1/2)^2+1/n)^{1/2}$ are in $C^1[0,1]$ and converge uniformly on $[0,1]$ to $|x-1/2|\notin C^1[0,1].$

Answer 2

It is well known that polynomials defined on $[0,1]$ are dense in $C[0,1]$ (with the uniform metric). Take any function of $C[0,1]-C^1[0,1]$ and approximate it by a sequence of polynomials; this is a sequence of $C^1[0,1]$ that is $\|\cdot\|_\infty$-Cauchy, but it converges to a function that does not belong in $C^1[0,1]$.