When reading about functional analysis I encountered the following example of a Banach space:
$ C^1 ([0,1])$ endowed with the norm $\|f\| = \|f\|_\infty + \|f'\|_\infty$.
where $\|\cdot\|_\infty$ denotes the $\sup$-norm.
At first it seemed to me that $C^1 ([0,1])$ endowed with the norm $\|f\| = \|f\|_\infty$ is also a Banach space. The norm $\|f\| = \|f\|_\infty + \|f'\|_\infty$ therefore seemed unnecessarily complicated. But I suspect this is not the case. Hence:
Could anyone provide an example of a Cauchy sequence w.r.t. $\|\cdot\|_\infty$ of $C^1$ functions such that the limit is not $C^1$?