Prove that $(C^1[0,1], \|\cdot\|)$ is not a Banach space

Take $\sqrt{x+1/n}$, which is $C^1[0,1]$ for each $n$ thanks to our shifting of the discontinuity in the derivative.

To see this converges uniformly, note that $\sqrt{x}$ is uniformly continuous on compact sets. On say $[0,2]$ we may use uniform continuity to meet any epsilon challenge with an $N$ not dependent on $x$ with $$ |\sqrt{x+1/N}-\sqrt{x}|<\epsilon $$