$I=[0,1]$. For $k \in \mathbb{N}$, denote by $C^k(I)$ the space of real-valued functions on $I$ possessing continuous derivatives up to order $k$ on $I$, including one-sided derivative at the end points $0$ and $1$. Define, for $p \in [1,\infty)$,
$$ \|f \| = \sum_{i=0}^k \|f^{(i)}\|_{L^p(I)} $$
where $f^{(i)}$ denotes $i$th derivative of $f \in C^k(I)$ and $\|f\|_{L^p}$ is $L^p$ - norm with Lebesgue measure.
I'm trying to show that the space $C^k(I)$ equipped with the norm $\| \cdot \|$ is not complete.
Since completeness means every cauchy sequence$\{f_n\}_{n=1}^\infty$ in $C^k(I)$ is a convergent sequence, I think I need to find some counter example that cauchy sequence is not convergent.
I couldn't find any counter example yet and I am curious about how could I think out it?